Computer and Robot Vision Icv2.csie.ntu.edu.tw/CV/_material/CV1_CH11_2019.pdf · Computer and Robot Vision I Chapter 11 Arc Extraction and Segmentation Presented by: 王柏川 Email:

Computer and Robot Vision I

Chapter 11

Arc Extraction and Segmentation

Presented by 王柏川

Email d08922016ntuedutw

Tel 0929621000

授課教授傅楸善博士

111 Introduction

In some image sets lines curves and

circular arcs are more useful than regions or

helpful in addition to regions

object recognition

stereo matching

document analysis

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111 Introduction

In this Ch we discuss techniques for

extracting sequences of pixels that belong to

the same curve

Source a segmented or

labeled image

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111 Introduction

Grouping Operation

After edge labeling or image segmentation

To extract sets or sequences of labeled or

border pixel positions

The extracted pixelrsquos positions

belong to the same curve (group)

sequence of pixels group features

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111 Introduction

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Labeling

edge detection label each pixel as edge or not

additional properties edge direction gradient

magnitude edge contrasthellip

Grouping

grouping operation edge pixels participating in

the same region boundary are grouped together

into a sequence

boundary sequence simple pieces

analytic descriptions shape-matching

112 Extracting Boundary Pixels

from a Segmented Image

Regions have been determined by

segmentation or connected components

boundary of each region can be extracted

Boundary extraction for small-sized images

1 scan through the image first border of

each region

2 first border of each region follow the

border of the connected component around in

a clockwise direction until reach itselfDC amp CV Lab

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memory problems for large-sized one simple boarder-tacking algo

may result in excessive IO to storage

4K UHD ndash (3840thinsptimesthinsp2160) ~= 83M pixel

Many techniques like down-sampling

It may be a problem back to 1992 how about TODAY

Border-tracking algorithm border1 Input symbolic image

2 Output a clockwise-ordered list of the coordinates of its border pixels

3 In one left-right top-bottom scan through the image

4 During execution

there are 3 sets of regions current past futureDC amp CV Lab

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1122 Border-Tracking Algorithm

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Past region null

Current region 2

Future region 1


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Past region 1

Current region 2

Future region null

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(active)


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(Inactive) CHAINSET (2) -gt (25) -gt (26)

(active) CHAINSET (1) -gt (32)

(32) NEIGHB (33) (42)


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- (33) NEIGHB (32)(34)(43)

- (active) CHAINSET (1) -gt (32) -gt (33)


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(Inactive) CHAINSET (2) -gt (25) -gt (2 6)

(active) CHAINSET (1) -gt (32) -gt (33)

- (34) NEIGHB (33)(44)

- (active) CHAINSET (1) -gt (32) -gt (33) -gt (34)


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(Inactive) CHAINSET (1) -gt (32) -gt (33) -gt (34)


(35) NEIGHB (25) (36) (45)


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(inactive) CHAINSET (2) -gt (25) -gt (26)

(inactive) CHAINSET (1) -gt (32) -gt (33) -gt (34)


- (36) NEIGHB (26) (35) (46)

- active CHAINSET change (clockwise-ordered)



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(42) NEIGHB (33)(43)(52)



(inactive) CHAINSET (2) -gt (35)


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helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip

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CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

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113 Linking One-Pixel-Wide

Edges or Lines

Border tracking each border bounded a closed

region NO point would be split into two or

more segments

Tracking symbolic edge (line) segments more

complex

value1 edge 0 non-edge

not necessary for edge pixels to bound closed regions

segments consist of connected edge pixels that go from

endpoint corner or junction to endpoint corner or junction

-gt pixeltype() is more complexDC amp CV Lab

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Edges or Lines

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INLIST OUTLIST


Edges or Lines

pixeltype() determines a pixel point

isolated point

starting point of an new segment

interior pixel of an old segment

ending point of an old segment

junction

corner

Instead of past current future regions there

are past current future segmentsDC amp CV Lab

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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114 Edge and Line Linking Using

Directional Information

Previous edge_track no directional

information

In this section assume each pixel is marked

to indicate whether it is an edge (line)

Yes there is an associated angular direction

pixels that have similar enough directions can form

connected chains and be identified as an arc segment

The arc has a good fit to a simple curvelike line

No

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The linking process -gt scan the labeled edge

image in top-down left-right and if a labeled

pixel has

No previous labeled neighbors

It begins a new group

One previous labeled neighbors

T-test

Two or more previous labeled neighbors

T-test

Merge testDC amp CV Lab

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initialize the scatter of group -gt

priori variance

of pixels

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If an encountered label pixel has previously

encountered labeled neighbors Measure t-

statistic

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If the pixel is added to the group

the mean and scatter of the group are

updated

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If there are two or more previously

encountered labeled neighbors

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then merge two groups

Create a new group having

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115 Segmentation of Arcs into

Simple Segments

Arc segmentation partition

extracted digital arc sequence digital arc

subsequences (each is a maximal sequence

that can fit a straight or curve line)

Simple arc segment straight-line or curved-

arc segment

The endpoints of the subsequences are

called corner points or dominant points

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Simple Segments

Identification of all locations

(a) have sufficiently high curvature

(b) are enclosed by different lines and curves

techniques iterative endpoint fitting and

splitting using tangent angle deflection or

high curvature as basis of the segmentation

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1151 Iterative Endpoint Fit and

Split

To segment a digital arc sequence into

subsequences that are sufficiently straight

one distance threshold d

L=(rc)| αr+βc+γ=0 where |(αβ)|=1 The straight line composed by 2 endpoints of this arc

di=| αri+βci+γ | |(αβ)| = | αri+βci+γ | The length of the perpendicular line composed by any single

pixel of this arc and L

dm=max(di)

If dmgt d then split at the point (rmcm)

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Split

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Split

L

dm=max(di)

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Split

L

dm=max(di)


Split

Circular arc sequence

initially split by two points apart in any direction

Sequence only composed of two line

segments

Golden section search

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1152 Tangential Angle Deflection

To identify the locations where two line

segments meet and form an angle

an(k)=(rn-k ndash rn cn-k - cn)

bn(k)=(rn ndash rn+k cn ndashcn+k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

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(rn-k ndash rn cn-k - cn)

(rn ndash rn+k cn ndashcn+k)

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an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

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At a place where two line segments meet

the angle will be larger cosθn(kn) smaller

A point at which two line segments meet

cosθn(kn) lt cosθi(ki) for all i|n-i| ≦ kn2

k=

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1153 Uniform Bounded-Error

Approximation

segment arc sequence into maximal pieces

whose points deviate le given amount

optimal algorithms excessive computational

complexity


Approximation

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1154 Breakpoint Optimization

after an initial segmentation shift breakpoints

to produce a better arc segmentation

first shift odd final point (ie even beginning

point) and see whether the max error is

reduced by the shift

If reduced then keep the shifted breakpoints

then shift even final point (ie odd beginning

point) and do the same things


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1155 Split and Merge

first split arc into segments with the error

sufficiently small

second merge successive segments if

resulting merged segment has sufficiently

small error

third try to adjust breakpoints to obtain a

better segmentation

repeat until all three steps produce no further

change


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1156 Isodata Segmentation

Iterative Self-organizing Data Analysis

Techniques Algorithm

iterative isodata line-fit clustering procedure

determines line-fit parameter

then each point assigned to cluster whose

line fit closest to the point


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1157 Curvature

The curvature is defined at a point of arc

length s along the curve by

Δs the change in arc length

Δθ the change in tangent angle

ss

s

0lim)(

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1157 Curvature

natural curve breaks curvature maxima and

minima

curvature passes through zero local shape

changes from convex to concave

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1157 Curvature

surface elliptic when limb in line drawing is

convex

surface hyperbolic when its limb is concave

surface parabolic wherever curvature of limb

zero

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1157 Curvature

Nalwa A Guided Tour of Computer Vision

Fig 414 1048576

116 Hough Transform

Hough Transform method for detecting

straight lines and curves on gray level

images

Hough Transform template matching

The Hough transform algorithm requires an

accumulator array whose dimension

corresponds to the number of unknown

parameters in the equation of the family of

curves being soughtDC amp CV Lab

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Finding Straight-Line Segments

Line equation y=mx+b

point (xy)

slope m intercept b

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x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

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x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

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x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

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x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

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x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1


Vertical lines m=infin doesnrsquot work

d perpendicular distance from line to origin

θ the angle the perpendicular makes with the

x-axis (column axis)

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(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

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cossin crd

r

c

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Example

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y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

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cossin crd

r

(11)

(01)

(10) (12)

(23)

c-45deg 0deg 45deg 90deg

(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

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accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

-45deg 0deg 45deg 90deg

(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

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accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

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accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


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DC amp CV Lab

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Finding Circles

row

column

row-coordinate of the center

column-coordinate of the center

radius

implicit equation for

a circle

Finding Circles

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Extensions

The Hough transform method can be

extended to any curve with analytic equation

of the form where denotes an

image point and is a vector of parameters

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0)( axf

x

a

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117 Line Fitting

points before noise perturbation

lie on the line

noisy observed value

independent and identically

distributed with mean 0 and variance 2

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117 Line Fitting

procedure for the least-squares fitting of line to

observed noisy values

principle of minimizing the squared residuals

under the constraint that

Lagrange multiplier form

1ˆˆ subject to ˆˆˆˆˆ min 22

1

22

N

n

nn )γcβrα(

1ˆˆ 22

N)γcβrα()γcβrα(N

n

nn

N

n

nn )1ˆˆ( - ˆˆˆˆˆ)]1ˆˆ(- ˆˆˆˆˆ[ 22

1

222

1

22

117 Line Fitting

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rarr

rarr

rarr

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rarr

rarr

rarr

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1172 Principal-Axis Curve Fit

The principal-axis curve fit is obviously a

generalization of the line-fitting idea

The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2

1 )()()( yyxfxyyxfxyxf

1)()()( 654 yxfyyxfxyxf

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The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1

2 1ˆ subject to )ˆˆ(

)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf

118 Region-of-Support

Determination

region of support too large fine features

smoothed out

region of support too small many corner

points or dominant points produced

k=

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Determination

Teh and Chin

Calculate until

Region of support

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119 Robust Line Fitting

Fit insensitive to a few outlier points

Give a least-squares formulation first and then

modify it to make it robust

1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11

In the weighted least-squares sense

otherwise 0

constantc ofmedian Z 1)( if ])(1[

22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

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1110 Least-Squares Curve

Fitting

Determine the parameters of the curve that

minimize the sum of the squared distances

between the noisy observed points and the

curve

0)( subject to )ˆ()ˆ( min 22

1

wcrfccrr nnnn

N

n

nn

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Fitting


1

wcrfccrr nnnn

N

n

nn

)ˆˆ( nn cr )( nn cr

0)( wcrf nn

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Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

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rarr

rarr

rarr

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Fitting

Distance d between and the curve


1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr

11101 Gradient Descent

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First-order iterative technique in minimization

problem

Initial value

(t+1)-th iteration

First-order Taylor series expansion around

should be in the negative gradient direction

0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

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11102 Newton Method

Second-order iterative technique in

minimization problem

Second-order Taylor series expansion around

second-order partial derivatives Hessian

Take partial derivatives to zero with respect to

tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

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11104 Fitting to a Circle

Circle

0)( wcrf

222 )()()()( RbcarRbacrfwcrf

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

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N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

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11106 Fitting to a Conic

In conic

1)())((2)()()( 22 bcCbcarBarACBAbafwcrf

)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

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11103 Second-Order

Approximation to Curve Fitting

==Nalwa A Guided Tour of Computer Vision

Fig 415== 1048576

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11109 Uniform Error Estimation


Fig 31 1048576

The End

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111 Introduction

In some image sets lines curves and

circular arcs are more useful than regions or

helpful in addition to regions

object recognition

stereo matching

document analysis

DC amp CV Lab

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111 Introduction



the same curve


labeled image

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111 Introduction

Grouping Operation







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111 Introduction

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Labeling




Grouping



into a sequence










each region




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4 During execution


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Past region null

Current region 2

Future region 1


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Past region 1

Current region 2

Future region null

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(active)


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(32) NEIGHB (33) (42)


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- (33) NEIGHB (32)(34)(43)



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- (34) NEIGHB (33)(44)



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(35) NEIGHB (25) (36) (45)


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- (36) NEIGHB (26) (35) (46)




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(42) NEIGHB (33)(43)(52)





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CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

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Edges or Lines



more segments


complex






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Edges or Lines

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INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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Edges or Lines

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information







No

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pixel has




T-test


T-test


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priori variance

of pixels

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statistic

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updated

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Simple Segments






arc segment



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Simple Segments







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Split







dm=max(di)


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Split

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Split

L

dm=max(di)

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Split

L

dm=max(di)


Split




segments


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DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

111 Introduction



the same curve


labeled image

DC amp CV Lab

CSIE NTU

111 Introduction

Grouping Operation







DC amp CV Lab

CSIE NTU

111 Introduction

DC amp CV Lab

CSIE NTU

Labeling




Grouping



into a sequence










each region




CSIE NTU












4 During execution


CSIE NTU


DC amp CV Lab

CSIE NTU

Past region null

Current region 2

Future region 1


DC amp CV Lab

CSIE NTU

Past region 1

Current region 2

Future region null

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(active)


DC amp CV Lab

CSIE NTU



(32) NEIGHB (33) (42)


DC amp CV Lab

CSIE NTU



- (33) NEIGHB (32)(34)(43)



DC amp CV Lab

CSIE NTU



- (34) NEIGHB (33)(44)



DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

111 Introduction

Grouping Operation







DC amp CV Lab

CSIE NTU

111 Introduction

DC amp CV Lab

CSIE NTU

Labeling




Grouping



into a sequence










each region




CSIE NTU












4 During execution


CSIE NTU


DC amp CV Lab

CSIE NTU

Past region null

Current region 2

Future region 1


DC amp CV Lab

CSIE NTU

Past region 1

Current region 2

Future region null

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(active)


DC amp CV Lab

CSIE NTU



(32) NEIGHB (33) (42)


DC amp CV Lab

CSIE NTU



- (33) NEIGHB (32)(34)(43)



DC amp CV Lab

CSIE NTU



- (34) NEIGHB (33)(44)



DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

111 Introduction

DC amp CV Lab

CSIE NTU

Labeling




Grouping



into a sequence










each region




CSIE NTU












4 During execution


CSIE NTU


DC amp CV Lab

CSIE NTU

Past region null

Current region 2

Future region 1


DC amp CV Lab

CSIE NTU

Past region 1

Current region 2

Future region null

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(active)


DC amp CV Lab

CSIE NTU



(32) NEIGHB (33) (42)


DC amp CV Lab

CSIE NTU



- (33) NEIGHB (32)(34)(43)



DC amp CV Lab

CSIE NTU



- (34) NEIGHB (33)(44)



DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU








each region




CSIE NTU












4 During execution


CSIE NTU


DC amp CV Lab

CSIE NTU

Past region null

Current region 2

Future region 1


DC amp CV Lab

CSIE NTU

Past region 1

Current region 2

Future region null

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(active)


DC amp CV Lab

CSIE NTU



(32) NEIGHB (33) (42)


DC amp CV Lab

CSIE NTU



- (33) NEIGHB (32)(34)(43)



DC amp CV Lab

CSIE NTU



- (34) NEIGHB (33)(44)



DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU












4 During execution


CSIE NTU


DC amp CV Lab

CSIE NTU

Past region null

Current region 2

Future region 1


DC amp CV Lab

CSIE NTU

Past region 1

Current region 2

Future region null

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(active)


DC amp CV Lab

CSIE NTU



(32) NEIGHB (33) (42)


DC amp CV Lab

CSIE NTU



- (33) NEIGHB (32)(34)(43)



DC amp CV Lab

CSIE NTU



- (34) NEIGHB (33)(44)



DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

Past region null

Current region 2

Future region 1


DC amp CV Lab

CSIE NTU

Past region 1

Current region 2

Future region null

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(active)


DC amp CV Lab

CSIE NTU



(32) NEIGHB (33) (42)


DC amp CV Lab

CSIE NTU



- (33) NEIGHB (32)(34)(43)



DC amp CV Lab

CSIE NTU



- (34) NEIGHB (33)(44)



DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

Past region 1

Current region 2

Future region null

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(active)


DC amp CV Lab

CSIE NTU



(32) NEIGHB (33) (42)


DC amp CV Lab

CSIE NTU



- (33) NEIGHB (32)(34)(43)



DC amp CV Lab

CSIE NTU



- (34) NEIGHB (33)(44)



DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(active)


DC amp CV Lab

CSIE NTU



(32) NEIGHB (33) (42)


DC amp CV Lab

CSIE NTU



- (33) NEIGHB (32)(34)(43)



DC amp CV Lab

CSIE NTU



- (34) NEIGHB (33)(44)



DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(active)


DC amp CV Lab

CSIE NTU



(32) NEIGHB (33) (42)


DC amp CV Lab

CSIE NTU



- (33) NEIGHB (32)(34)(43)



DC amp CV Lab

CSIE NTU



- (34) NEIGHB (33)(44)



DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(active)


DC amp CV Lab

CSIE NTU



(32) NEIGHB (33) (42)


DC amp CV Lab

CSIE NTU



- (33) NEIGHB (32)(34)(43)



DC amp CV Lab

CSIE NTU



- (34) NEIGHB (33)(44)



DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(active)


DC amp CV Lab

CSIE NTU



(32) NEIGHB (33) (42)


DC amp CV Lab

CSIE NTU



- (33) NEIGHB (32)(34)(43)



DC amp CV Lab

CSIE NTU



- (34) NEIGHB (33)(44)



DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU



(32) NEIGHB (33) (42)


DC amp CV Lab

CSIE NTU



- (33) NEIGHB (32)(34)(43)



DC amp CV Lab

CSIE NTU



- (34) NEIGHB (33)(44)



DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU



- (33) NEIGHB (32)(34)(43)



DC amp CV Lab

CSIE NTU



- (34) NEIGHB (33)(44)



DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU



- (34) NEIGHB (33)(44)



DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




(35) NEIGHB (25) (36) (45)


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




- (36) NEIGHB (26) (35) (46)




DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


(42) NEIGHB (33)(43)(52)





DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU







DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


CHAINSET

(1)(32)(33)(34)(44)(54)

(1)(42)(52)(53)

(2)(25)(26)(36)(46)(56)(66)

(2)(35)(45)(55)(65)

CHAINSET

(1)(32)(33)(34)(44)(54)(53)(52)(42)

(2)(25)(26)(36)(46)(56)(66)(65)(55)

(45)(35)

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines



more segments


complex






CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU

INLIST OUTLIST


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines


isolated point




junction

corner



CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Edges or Lines

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU




information







No

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU





pixel has




T-test


T-test


CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU





priori variance

of pixels

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU





statistic

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU





updated

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU



DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU





DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Simple Segments






arc segment



DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Simple Segments







DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Split







dm=max(di)


DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Split

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Split

L

dm=max(di)


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Split




segments


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU




DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU




an(k)

bn(k)

||)(|| ||)(||

)()()](cos[

kbka

kbkak

nn

T

nnn

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU






k=

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Approximation




complexity


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Approximation

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU











DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



sufficiently small



small error


better segmentation


change


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU









DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature





ss

s

0lim)(

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature


minima



DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature


convex



zero

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

1157 Curvature


Fig 414 1048576

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

116 Hough Transform



images







CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(11)



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU



point (xy)

slope m intercept b

DC amp CV Lab

CSIE NTU

x

y

b

m

1=m+b

(22)

(11)

2=2m+b

(10)

y=1x+0

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

x

y

b

m

(11)

(11)

(10)

(01) (21)

(32)

(10)

(21)

(32)

(01)

(01)

(1-1)

y=1

y=x-1






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU






DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

(dsinθdcosθ)

(rc)

cossin crd

ΘΘ

Θ

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

c

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

y

(11)

(10)

(01) (21)

(32)

cossin crd x

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

cossin crd

r

(11)

(01)

(10) (12)

(23)


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -


(01) 0707 1 0707 0

(10) -0707 0 0707 1

(11) 0 1 1414 1

(12) 0707 2 2121 1

(23) 0707 3 3535 2

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

Example

DC amp CV Lab

CSIE NTU

accumulator array

-0707 0 0707 1 1414 2 2121 3 3535

-45deg 1 1 3 - - - - - -

0deg - 1 - 2 - 1 - 1 -

45deg - - 2 - 1 - 1 - 1

90deg - 1 - 3 - 1 - - -

r

(11)

(01)

(10) (12)

(23)

c


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

Finding Circles

row

column



radius


a circle

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

Finding Circles

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

Extensions





DC amp CV Lab

CSIE NTU

0)( axf

x

a

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

117 Line Fitting


lie on the line




DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

117 Line Fitting







1

22

N

n

nn )γcβrα(

1ˆˆ 22


n

nn

N

n


1

222

1

22

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

117 Line Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU




The curve

eg conics

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

022 feydxcybxyax2

32

2



DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


The curve

minimize

K

1k

2

1

1 )()(0 k

K

k

kk crfcrf

KN

n

nn crf1k

2

k

1


)1()ˆˆ( min1

2

1

2

K

k

kn

N

n

n crf

)1(])([1

2

1 1

K

k

k

N

n

K

k

nnkk crf


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Determination


smoothed out



k=

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Determination

Teh and Chin

Calculate until

Region of support

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU





1 subject to

ˆ

ˆ

ˆ2

minarg

pp t

p

pA

p

cr

cr

cr

A

NxNN

ˆ

ˆ

ˆ

1ˆˆ

1ˆˆ

1ˆˆ

3

22

11


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


otherwise 0


22

1

1

iii

i

N

k

ecZ

e

cZ

e

w

w

w

W


1 subject to

ˆ

ˆ

ˆ

minarg

pWWp t

p

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Fitting




curve


1

wcrfccrr nnnn

N

n

nn

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Fitting


1

wcrfccrr nnnn

N

n

nn


0)( wcrf nn

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Fitting

N

n

nn wcrf1

2)ˆˆ( min


0)( wcrf nn


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


Fitting

DC amp CV Lab

CSIE NTU

rarr

rarr

rarr

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Fitting



1

wcrfccrr nnnn

N

n

nn

22

)ˆˆ()ˆˆ(

)ˆˆ(

wcrfc

wcrfr

wcrfd

nnnn

nn

N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)ˆˆ( nn cr


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU


DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU



problem

Initial value

(t+1)-th iteration



0w

www tt

1

)()()()()( 222

1

2

t

T

ttt wwwwww

tw

w

22

2

22

)(

)()(

t

tt

w

www

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

11102 Newton Method






tw

wHwwwwww T

t

T

tt

)(

2

1)()()()( 222

w

)( 0)()( 212

tt wHwwwH

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


Circle

0)( wcrf


N

n

nnnn

nn

wcrfc

wcrfr

wcrfw

122

22

)ˆˆ()ˆˆ(

)ˆˆ()( min

)(2 )(2 bcfc

arfr

N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


N

n nn

nn

bcar

RbcarRba

122

22222

])()ˆ[(4

])()ˆ[()( min

N

n

nn RbcarRba1

2222 ])()ˆ([)( min

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU


In conic


)(2)(2

)(2)(2

bcCarBfc

bcBarAfr

N

n nnnn

nnnn

bcCarBbcBarA

bcCbcarBarA

122

2222

)]ˆ()ˆ([)]ˆ()ˆ([

]1)ˆ()ˆ)(ˆ(2)ˆ([

4

1

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU

11103 Second-Order



Fig 415== 1048576

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

DC amp CV Lab

CSIE NTU



Fig 31 1048576

The End

DC amp CV Lab

CSIE NTU

The End

DC amp CV Lab

CSIE NTU

Documents

Computer and Robot Vision Icv2.csie.ntu.edu.tw/CV/_material/CV1_CH11_2019.pdf · Computer and Robot Vision I Chapter 11 Arc Extraction and Segmentation Presented by: 王柏川 Email: