Lec-5 Image Analysis-Boundary Detection

8/10/2019 Lec-5 Image Analysis-Boundary Detection

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Image Analysis

Boundary Detection

Dr. Ziya Telatar


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Image Analysis

Feature Extraction

- Spatial Features

- Transform Features

- Edges and

Boundaries- Shape Features

- Moments

- Texture

Segmentation

- Template Matching

- Thresholding

- Boundary

Detection

- Clustering

- Quad-Trees

- Texture matching

Classification

- Clustering

- Statistical

- Decision Trees

- Similarity Measures

- Min. Spanning

Trees

IMAGE ANALYSIS TECHNIQUES


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Image Analysis Preview

Feature extraxtion is the first and importantstep for succesfully analysis

Segmentation is to subdivide an image into

its constituent regions or objects. Segmentation should stop when the objectsof interest in an application have beenisolated.

Classification is closely related tosegmentation


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Principal approaches

Segmentation based feature extractionalgorithms generally are based on one of 2

basis properties of intensity values discontinuity : to partition an image based on

abrupt changes in intensity (such as edges)

similarity : to partition an image into regions

that are similar according to a setof predefined criteria.


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Detection of Discontinuities

detect the three basic types of gray-level

discontinuities

points , lines , edges

the common way is to run a mask through

the image


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Point Detection

a point has been detected at the location on

which the mark is centered if

|R| T

where

T is a nonnegative threshold

R is the sum of products of the coefficients with

the gray levels contained in the region

encompassed by the mark.


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Point Detection

Note that the mark is the same as the mask

of Laplacian Operation

The only differences that are considered of

interest are those large enough (as

determined by T) to be considered isolated

points.|R| T


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Example


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Line Detection

Horizontal mask will result with max response when a linepassed through the middle row of the mask with aconstant background.

the similar idea is used with other masks.

note: the preferred direction of each mask is weighted witha larger coefficient (i.e.,2) than other possible directions.


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Line Detection

Apply every masks on the image

let R1, R2, R3, R4 denotes the response of

the horizontal, +45 degree, vertical and -45degree masks, respectively.

if, at a certain point in the image

|Ri| > |Rj|, for all ji, that point is said to be more likely

associated with a line in the direction of maski.


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Line Detection

Alternatively, if we are interested in detectingall lines in an image in the direction defined

by a given mask, we simply run the maskthrough the image and threshold the absolutevalue of the result.

The points that are left are the strongestresponses, which, for lines one pixel thick,correspond closest to the direction defined bythe mask.


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Example


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Edge Detection

the most common approach for detecting

meaningful discontinuities in gray level.

we discuss approaches for implementing

first-order derivative (Gradient operator)

second-order derivative (Laplacian operator)

Here, we will talk only about their properties

for edge detection.


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Basic Formulation

an edge is a set of connected pixels that

lie on the boundary between two regions.

an edge is a local concept whereas a

region boundary, owing to the way it is

defined, is a more global idea.


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Ideal and Ramp Edges

because of optics,sampling, imageacquisitionimperfection


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Thick edge

The slope of the ramp is inversely proportional to thedegree of blurring in the edge.

We no longer have a thin (one pixel thick) path.

Instead, an edge point now is any point contained in theramp, and an edge would then be a set of such points thatare connected.

The thickness is determined by the length of the ramp.

The length is determined by the slope, which is in turndetermined by the degree of blurring.

Blurred edges tend to be thick and sharp edges tendto be thin


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First and Second derivatives

the signs of the derivativeswould be reversed for an edgethat transitions from light to

dark


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Second derivatives

produces 2 values for every edge in animage (an undesirable feature)

an imaginary straight line joining the

extreme positive and negative values ofthe second derivative would cross zeronear the midpoint of the edge. (zero-crossing property)

quite useful for locating the centers of thickedges


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Noisy Images

First column: images andgray-level profiles of aramp edge corrupted byrandom Gaussian noiseof mean 0 and = 0.0,

0.1, 1.0 and 10.0,respectively.

Second column: first-derivative images andgray-level profiles.

Third column : second-derivative images andgray-level profiles.


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Keep in mind

fairly little noise can have such a

significant impact on the two key

derivatives used for edge detection in

images

image smoothing should be serious

consideration prior to the use of

derivatives in applications where noise islikely to be present.


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Edge point

to determine a point as an edge point

the transition in grey level associated with thepoint has to be significantly stronger than the

background at that point. use threshold to determine whether a value is

significant or not.

the points two-dimensional first-order

derivative must be greater than a specifiedthreshold.

f


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Gradient Operator

first derivatives are implemented using

the magnitude of the gradient.

y

fx

f

G

G

y

xf

21

22

21

22 ][)f(

yf

xf

GGmagf yx

the magnitude becomes nonlinearyx GGf

commonly approx.


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Gradient direction

Let (x,y) represent the direction angle of

the vector f at (x,y)

(x,y) = tan-1(Gy/Gx)

The direction of an edge at (x,y) is

perpendicular to the direction of the

gradient vector at that point


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Gradient Masks


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Diagonal edges with Prewitt

and Sobel masks

Sobel masks haveslightly superiornoise-suppression

characteristics whichis an important issuewhen dealing withderivatives.


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Example


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Example


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Example


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Laplacian

2

2

2

22 ),(),(

y

yxf

x

yxff

(linear operator)

Laplacian operator

)],(4)1,()1,(

),1(),1([2

yxfyxfyxf

yxfyxff


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Laplacian of Gaussian

Laplacian combined with smoothing as a

precursor to find edges via zero-crossing.

2

2

2

)(

r

erh

where r2= x2+y2, andis the standard deviation

2

2

24

222

)(

r

er

rh


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Mexican hat

the coefficient must be sum to zero

positive central term

surrounded by an adjacent negative region (a function of distance)zero outer region


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Linear Operation

second derivation is a linear operation

thus, 2f is the same as convolving the

image with Gaussian smoothing function

first and then computing the Laplacian of

the result


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Example

a). Original imageb). Sobel Gradient

c). Spatial Gaussiansmoothing functiond). Laplacian maske). LoG

f). Threshold LoGg). Zero crossing


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Zero crossing & LoG

Approximate the zero crossing from LoGimage

to threshold the LoG image by setting all

its positive values to white and all negativevalues to black.

the zero crossing occur between positive

and negative values of the thresholdedLoG.


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Zero crossing vs. Gradient

Attractive

Zero crossing produces thinner edges

Noise reduction

Drawbacks Zero crossing creates closed loops. (spaghetti

effect)

sophisticated computation.

Gradient is more frequently used.


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Edge Linking and Grouping

Define what it means for a group to be

good.

Usually this involves simplifications

Search for the best group.

Usually this is intractable, so short-cuts are

needed.


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Parametric Grouping: Grouping

Points into Lines

Basic Facts about Lines

(a,b)

c (x,y) is on line if (x,y).(a,b) = cax + by = c

Distance from (x,y) to line is

(a,b).*(x,y) = ax + by


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Line Grouping Problem


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This is difficult because of:

Extraneous data: Clutter

Missing data

Noise


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RANSAC: Random Sample

Consensus

Generate a bunch of reasonable

hypotheses.

Test to see which is the best.


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RANSAC for Lines Generate Lines using Pairs of Points

How many samples?

Supposep is fraction of points from line.

npoints needed to define hypothesis (2 for lines)

k samples chosen.

Probability one sample correct is:

kn

p)1(1


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RANSAC for Lines: Continued

Decide how good a line is:

Count number of points within eof line.

Parameteremeasures the amount of noise

expected.

Other possibilities. For example, for these

points, also look at how far they are.

Pick the best line.


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Some comparisons

Complexity of RANSAC n*n*n

Complexity of Hough n*d

Error behavior: both can have problems,

RANSAC perhaps easier to understand. Clutter: RANSAC very robust, Hough falls

apart at some point.

There are endless variations that improvesome of Houghs problems.


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Boundary Extraction

Boundaries are linked edges that characterize

the shape of an object

Connectivity Boundaries can be found by tracing the connected

edges

On a rectangular grid, a pixel is said to be 4- or 8-

connected when it has the same properties as one of

its nearest 4 or 8 neighbors, respectively


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Neighbors of Pixels

4- Neighborhood

(x+1,y) (x-1,y) (x,y+1) (x,y-1)

8- Neighbor with diagonal pixels

(x+1,y+1) (x+1,y-1) (x-1,y+1) (x-1,y-1)


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Boundary Extraction

Contour Following

Contour-following algorithms trace boundaries by

ordering successive edge points.

Some example of algorithms are:

Edge Linking and Hueristic Graph Search Dynamic programming

Hough Transform


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Boundary Extraction

Edge Linking:

edge detection algorithm are followed by

linking procedures to assemble edge pixels

into meaningful edges.

Basic approaches

Local Processing

Global Processing via the Hough Transform Global Processing via Graph-Theoretic

Techniques


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Boundary Extraction

Local Processing

analyze the characteristics of pixels in a small

neighborhood (say, 3x3, 5x5) about every

edge pixels (x,y) in an image. all points that are similar according to a set of

predefined criteria are linked, forming an edge

of pixels that share those criteria.


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Boundary ExtractionLocal Processing

Criteria

1. the strength of the response of thegradient operator used to produce the

edge pixel an edge pixel with coordinates (x0,y0) in a

predefined neighborhood of (x,y) is similar inmagnitude to the pixel at (x,y) if

|f(x,y) - f (x0,y0) | E


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Boundary ExtractionLocal Processing

Criteria

2. the direction of the gradient vector

an edge pixel with coordinates (x0,y0) in a

predefined neighborhood of (x,y) is similar in

angle to the pixel at (x,y) if

|(x,y) - (x0,y0) | < A


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Criteria

A point in the predefined neighborhood of (x,y) is

linked to the pixel at (x,y) if both magnitude

and direction criteria are satified.

the process is repeated at every location in theimage

a record must be kept

simply by assigning a different gray level to each

set of linked edge pixels.

fi d t l h i k


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Examplefind rectangles whose sizes makesthem suitable candidates for licenseplates

use horizontal andvertical Sobel operators

link conditions:gradient value > 25gradient direction differs < 15

eliminate isolated shortsegments

B d E t ti


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Boundary ExtractionHough Transformation (Line)

yi=axi+ b b = - axi+ yi

ab-plane or parameter space

xy-plane

all points (xi,yi) contained on the same line must have lines in

parameter space that intersect at (a,b

)


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Accumulator cells

(amax, amin) and (bmax, bmin) are

the expected ranges of slope

and intercept values.

all are initialized to zero

if a choice of apresults in

solution bqthen we let

A(p,q) = A(p,q)+1

at the end of the procedure,

value Q in A(i,j) corresponds to

Q points in the xy-plane lying

on the line y = aix+bj b = - axi+ yi


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-plane

problem of using equation y = ax + b is that value of a is infinite for avertical line.

To avoid the problem, use equation x cos + y sin = to represent aline instead.

vertical line has = 90with equals to the positive y-intercept or =

-90with equals to the negative y-intercept

x cos + y sin =

-plane

= 90 measured with respect to x-axis


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D2ofrange

where D is the distancebetween corners in the

image


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Generalized Hough Transformation

can be used for any function of the form

g(v,c) = 0

v is a vector of coordinates

c is a vector of coefficients


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Hough Transformation (Circle)

equation:

(x-c1)2+ (y-c2)

2= c32

three parameters (c1, c2, c3)

cube like cells accumulators of the form A(i, j, k)

increment c1and c2, solve of c3that satisfies theequation

update the accumulator corresponding to thecell associated with triplet (c1, c2, c3)

Edge linking based on Hough


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Edge-linking based on Hough

Transformation

1. Compute the gradient of an image andthreshold it to obtain a binary image.

2. Specify subdivisions in the -plane.

3. Examine the counts of the accumulatorcells for high pixel concentrations.

4. Examine the relationship (principally for

continuity) between pixels in a chosencell.


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The Hough Transform for

Lines A line is the set of points (x, y) such that

Different choices of , d>0 give different lines

For any (x, y) there is a one parameter familyof lines through this point. Just let (x,y) be

constants and , d be unknowns.

Each point gets to vote for each line in the

family; if there is a line that has lots of votes,

that should be the line passing through the

points


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Mechanics of the Hough

transform Construct an arrayrepresenting , d

For each point, render

the curve (, d) into this

array, adding one at

each cell

Difficulties

how big should the cells

be? (too big, and wecannot distinguish

between quite different

lines; too small, and

noise causes lines to be

missed)

How many lines?

count the peaks in the

Hough array

Who belongs to whichline?

tag the votes

Can modify voting, peak

finding to reflect noise. Big problem if noise in

Hough space different

from noise in image

space.


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Continuity

based on computing the distance between

disconnected pixels identified during

traversal of the set of pixels corresponding

to a given accumulator cell.

a gap at any point is significant if the

distance between that point and its closetneighbor exceeds a certain threshold.


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link criteria:1). the pixels belonged to one of the set of pixelslinked according to the highest count2). no gaps were longer than 5 pixels


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Boundary Representation

Proper representation of object boundaries isimportant for analysis and synthesis of shape

Shape analysis is often required for detection

and recognition of objects in a scene.

Shape synthesis is useful in computer-aideddesign (CAD) of parts and assemblies, image

simulation applications such as video games,cartoon movies, environmental modeling ofaircraft-landing testing and training, etc.

Methods for Boundary Representations


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Methods for Boundary Representations

Chain Codes The direction vectors between successive boundary pixels are encoded.

Fitting Line Segments Staright-line segments give simple approximation of curve boundaries.

Approximate the curve by the line segment joining its end points.

If the distance from the farthest curve point to the segment is greater than apredetermined quantitiy, join the points to line ends.

B-Spline Representation Piecewise polynomial functions that can provide local approximations of contours

of shapes using a small number of parameters.

B-Splines are used in shape synthesis and analysis, computer graphics, andrecognition parts from boundaries.

Fourier Descriptors Effect of Geometric Transformations

Boundary matching from FD

Autoregressive Models For arbitrary class of object boundaries, we have an ensemble of boundaries that

could be represented by a stochastic model.


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Example:

Boundary superimposed


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Region Representation

The shape of an object may be directly

represented by the region it occupies

Run-length codes

Quad-trees

Projections


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Moment Representation

The theory of moments provides an interestingand sometimes useful alternative to serious

expansions for representing shape of objects

Moment representation theorem Moment matching

Orthogonal moments

Moment invariants

Translation Scaling

Rotation and Reflection


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Structure

In many computer vision applications, the objects in ascene can be characterized satisfactorily by structurescomposed of line or arc patterns like handwritten orprinted characters, circuit diagrams and engineeringdrawings, etc.

Transformations useful for analysis of structure ofpatterns Medial Axes Transforms

Skeleton Algorithms Thinning Algorithms

Morphological Transforms


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Shape Features

The shape of an object refers to its profile

and physical structure

These characteristics can be represented

by the boundary, region, moment, and

structural representations


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Shape Features

Shape RepresentationRegenerative Features

- Boundaries

- Regions

- Moments- Structural and Syntactic

Measurement Features

Geometry

- Perimetry

- Area

- Max-min Radii

and eccentricity

- Corners

- Roundness

- Bending Energy

- Holes

- Euler Numbers

- Symmetry

Moments

- Center of Mass

- Orintation

- Bounding

Rectangle

- Best-Fit Ellipse

- Eccentricity


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Texture

Texture is observed in the structural patterns of objectssuch as wood, grain, sand, grass, and cloth

A texture contains several pixels, whose placementcould be periodic, quasi-periodic or random.

Natural textures are generally random

Artificial textures are often deterministic or periodic

Texture are classified into two main categories:Statistical and Structural.


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TextureClassification of Texture

Statistical

- ACF (Autocorrelation

Function)

- Transforms

- Edge-ness

- Concurrence Matrix

- TextureTransforms

- Random Field

Models

Other

Random

- Edge Density

- Extreme Density- Run Lengths

Mosaic Models

Structural

Periodic

- Primitives:

- Gray Levels- Shape

- Homogeneity

- Placement Rules

- Period

- Adjacency

- Closest

Distances


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Scene Matching and Detection

A problem of much significance in imageanalysis is the detection of change or presenceof an object in a given scene

Methods Image Subtraction

Template Matching and Area Correlation

Matched Filtering

Direct serch Two Dimensional Logarithmic search Sequential Search

Hiearchical Search

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Lec-5 Image Analysis-Boundary Detection