3-D Computer Vision CSc 83020

3-D Computer Vision CSc 83020 – Ioannis Stamos3-D Computer Vision CSc 83020 – Ioannis Stamos

3-D Computer Vision3-D Computer Vision CSc 83020 CSc 83020

Clustering methods and boundary Clustering methods and boundary representationsrepresentations


Image SegmentationImage Segmentation Generate clusters (regions) of pixels Generate clusters (regions) of pixels

that correspond to meaningful entities.that correspond to meaningful entities. Use metrics of “closeness” between Use metrics of “closeness” between

values.values. Use algorithms for combining “close” Use algorithms for combining “close”

values.values. Apply other constraints (connectivity).Apply other constraints (connectivity).


Example (Forsyth & Ponce)Example (Forsyth & Ponce)


Simple Clustering MethodsSimple Clustering Methods

Divisive clusteringDivisive clustering Everything is a big cluster at beginning Everything is a big cluster at beginning Split recursivelySplit recursively

Agglomerative clusteringAgglomerative clustering Each data (pixel) is a clusterEach data (pixel) is a cluster MergeMerge


Simple clusteringSimple clustering



What is a good inter-cluster What is a good inter-cluster distance?distance?

How many clusters are there?How many clusters are there?




Clustering by K-meansClustering by K-means

Assume that the number of clusters (k) is known.Assume that the number of clusters (k) is known. Each cluster has a center CEach cluster has a center Ci i (i=1..k)(i=1..k) Each data-point is a vector Each data-point is a vector xxj j (j=1..Number of pixels)(j=1..Number of pixels)

Examples:Examples: xj=[x-coord, y-coord, gray-value]xj=[x-coord, y-coord, gray-value]

or xj=[gray-value]or xj=[gray-value]or xj=[red-value, green-value, blue-value]or xj=[red-value, green-value, blue-value]

Assume that elements are close to center of clusters.Assume that elements are close to center of clusters. Minimize:Minimize:


Clustering by K-meansClustering by K-means Iterative algorithm:Iterative algorithm:

Allocate each point to center of closest Allocate each point to center of closest cluster (assuming centers are known)cluster (assuming centers are known)

Calculate centers of clusters (assuming Calculate centers of clusters (assuming allocations are known)allocations are known)

How do we start?How do we start?


Clustering by K-meansClustering by K-means








RANSACRANSAC

RANRANdom dom SASAmple mple CConsensusonsensus Model fitting methodModel fitting method Line-fitting exampleLine-fitting example

Fitting a line to a set of edges with 50% Fitting a line to a set of edges with 50% outliersoutliers

Least squares would failLeast squares would fail Solution: M-estimator or RANSACSolution: M-estimator or RANSAC


RANSAC (line-fitting RANSAC (line-fitting example)example)

Two edges (wout normal) define a line.Two edges (wout normal) define a line. General idea:General idea:

Pick two points.Pick two points. Write the equation of the line.Write the equation of the line. Check how many other points are “close” to line.Check how many other points are “close” to line. If number of “close” points is above threshold, doneIf number of “close” points is above threshold, done Otherwise, pick two new points.Otherwise, pick two new points.

Questions:Questions: Which points to pick?Which points to pick? Complexity in worst case?Complexity in worst case?





How large should k (max. number of iterations) How large should k (max. number of iterations) be?be?



How large should k (max. number of iterations) be?How large should k (max. number of iterations) be? Assume that Assume that w w is the probability of picking a “correct” is the probability of picking a “correct”

point (i.e. a point on the line).point (i.e. a point on the line). Since we are picking Since we are picking n n (=2 for lines) points, 1-(=2 for lines) points, 1-wwn n is the is the

probability of picking probability of picking nn “wrong” points. “wrong” points. If we iterate k times we want the probability of failure If we iterate k times we want the probability of failure

to be small: i.e.to be small: i.e. (1-(1-wwnn))k k = z => k = log(z)/log(1-= z => k = log(z)/log(1-wwnn))

If z=0.1 and w=0.5 then k=8 (n=2)If z=0.1 and w=0.5 then k=8 (n=2) If z=0.01 and w=0.5 then k=16 (n=2)If z=0.01 and w=0.5 then k=16 (n=2) If z=0.001 and w=0.1 then k = 687 (n=2)If z=0.001 and w=0.1 then k = 687 (n=2)

How is the formula affected by n?How is the formula affected by n?


RANSAC (Conclusions)RANSAC (Conclusions) When can this method be successful?When can this method be successful? Can we detect circles?Can we detect circles?

In that case how many points do you need to fit a In that case how many points do you need to fit a circle?circle?

Can we detect other shapes?Can we detect other shapes?


Boundary representation of Boundary representation of regionsregions


Representation of 2-D Representation of 2-D Geometric StructuresGeometric Structures

To MATCH image boundary/region with MODEL

boundary/region, they must represented in the same

manner.

•Boundary Representation

•Snakes – Extraction of arbitrary contours from image.

•Region Representation


Representation IssuesRepresentation Issues

Compact: Easy to Store & Match.Compact: Easy to Store & Match. Easy to manipulate & compute Easy to manipulate & compute

properties.properties. Captures Object/Model shape.Captures Object/Model shape. Computationally efficient.Computationally efficient.


Boundary RepresentationBoundary Representation

Polylines: concatenation of line segments.

Breakpoint

Matching on the basis of:# of line segmentslengths of line segmentsangle between consecutive segments


Running Least Squares Running Least Squares MethodMethod

ei

Move along the boundaryAt each point find line that fits previous points

(Least Squares)Compute the fit error E=Sum(ei) using previous pointsIf E exceeds threshold, declare breakpoint and start

a new running line fit.


Approximating Curves with Approximating Curves with PolylinesPolylines

•Draw Straight line between end-points of curve•For every curve point find distance to line.•If distance is less than tolerance level for all points, Exit•Else, pick point that is farthest away and use as breakpoint.

Introduce new segments.•Recursively apply algorithm to new segments.


Ψ-s CurveΨ-s Curve

s ψ

x

y

s

Ψ

1 2

3 4

5 6

x

y

12

3

4

5

6

π

2π


Ψ-s CurveΨ-s Curve

s ψ

x

y

s

Ψ

1 2

3 4

5 6

x

y

12

3

4

5

6

Ψ-s is periodic (2π)Horizontal section in Ψ-s curve

=> straight line in the ImageNon horizontal section in Ψ-s

=> arc in Image

π

2π


Slope-Density FunctionSlope-Density Function

s ψ

x

y H(Ψ)

Ψ

π 2π

HISTOGRAMLines

Arcs


Slope-Density FunctionSlope-Density Function

s ψ

x

y H(Ψ)

Ψ

π 2π

HISTOGRAM

H(Ψ) shifts as objects rotates.H(Ψ) wraps around

Lines

Arcs

Fourier DescriptorsFourier Descriptors

s ψ

x

y

Find: Ψ(s)Define:Φ(s)= Ψ(s)-(2πs)/P

P: Perimeter.2π: Period of Φ(s)

Φ(s) is a Continuous, Periodic function.Fourier Series for Periodic Functions:

Fourier Coefficients:

Pes

k

sikk

2,)( 0

1

0

P

sikk dsesP 0

0)(1

Φk’s capture shape informationMatch shapes by matching Φk’sUse finite number of Φk’s

Fourier DescriptorsInput Shape Power Spectrum Reconstruction

# of coefficients

Φk

Note: Reconstructed shapes are often not closed sinceonly a finite # of Φk’s are used.

1

0

)()(N

iii sBvsX

B-SplinesB-Splines•Piecewise continuous polynomials used to INTERPOLATE between Data Points.•Smooth, Flexible, Accurate.

s=0

s=1s=2

x0

x1x2

xi

x

s

Spline X(s)

Data Point

We want to find X(s) from points xiCubic Polynomials are popular:

BASIS FUNCTIONS

COEFFICIENTS

N

B-SplinesB-SplinesBi(s) has limited support (4 spans)

i-2 i-1 i i+1 i+2

Each span (i->i+1) has only 4 non-zero Basis Functions:Bi-1(s), Bi(s), Bi+1(s), Bi+2(s)

i i+1

Bi(s)

Bi-1(s) Bi+2(s)

Bi+1(s)

t=0 t=1t

s:

s:


B-SplinesB-Splines

i i+1

Bi(s)

Bi-1(s) Bi+2(s)

Bi+1(s)

t=0 t=1t

CubicPolynomials

1/6

4/6


B-SplinesB-Splines

i i+1

Bi(s)

Bi-1(s) Bi+2(s)

Bi+1(s)

t=0 t=1t

6/)()(

6/)1333()()(

6/)463()()(

6/)133()()(

332

2321

231

2301

ttCsB

ttttCsB

tttCsB

ttttCsB

i

i

i

i

CubicPolynomials

1/6

4/6


B-SplinesB-Splines

0141

0303

0363

1331

6

1

1)(

)()()()()(

21123

2312110

C

vvvvCttttx

vtCvtCvtCvtCtxT

iiiii

iiiii

If we compute v0, … vN => continuous representation for the curve.

B-SplinesB-Splines

)0(

...

)0()0()0()0(...............................)0(

)0()0()0()0(................)0(

)0()0()0()0()0(

4332211022

3322110011

2312011000

NN xx

vCvCvCvCxx

vCvCvCvCxx

vCvCvCvCxx

NN v

v

v

v

x

x

x

x

...

410......0

0............0

0...1410

0...0141

0...0014

6

1

...2

1

0

2

1

0

We have our N+1 data points:

And in matrix form:

We can solve for the vi’s from the xi’sBoundary condition for closed curves: v0=vN, v1=vN+1.

B-SplinesB-Splines

Tiiiii vvvvCttttx 21123 1)(

So, for any i we can find:

xi xi+1

xi(t)

t

t=0 t=1

Note: Local support.Spline passes through all data points xi.

B-Spline demo: http://www.doc.ic.ac.uk/~dfg/AndysSplineTutorial/BSplines.html


SnakesSnakesElastic band of arbitrary shape.Located near the image contour.


Snakes – ImplementationSnakes – Implementation

How can we minimize

i

imageicurviconti EEE

Discrete version: sum over the points.

Greedy minimization:For each point compute the best move over a small area.



How can we minimize

i




Set βi=0 for points of high curvature (corners)



How can we minimize

i




Set βi=0 for points of high curvature (corners)

Stop when a user-specified fraction of points does not move.


Synthetic Results

Real Experiment

Snakes demo: http://www.markschulze.net/snakes/


Region RepresentationRegion Representation

1 11 1 1 1 1

1 1 1 1 11 1 1 1

1

Spatial Occupancy Array

•Easy to implement.•Large Storage area.•Can apply set operations (unite/intersect).•Expensive for matching!


Quad TreesEfficient encoding ofSpatial Occupancy Arrayusing Resolution Pyramids.

Black: Fully Occupied.White: Empty.Gray: Partially Occupied.


Quad TreeLevel 0Level 1

Level 2

Level 3

Quad Tree Generation:Start with level 0. If Black or White, Terminate. Else

declare Gray node & expand with four sons.

For each son repeat above step.

NW NE

SW SE

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3-D Computer Vision CSc 83020