© 2002-2003 by yu hen hu 1 ece533 digital image processing image segmentation

© 2002-2003 by Yu Hen Hu1ECE533 Digital Image Processing

Image Segmentation


What is Image Segmentation?

Segmentation: » Split or separate an image into

regions» To facilitate recognition,

understanding, and region of interests (ROI) processing

Ill-defined problem» The definition of a region is context-

dependent

20 40 60

20

40

60


Outline

Discontinuity Detection» Point, edge, line

Edge Linking and boundary detection Thresholding Region based segmentation Segmentation by morphological

watersheds Motion segmentation


Point Detection

Apply detection mask, followed by threshold detection


Line Detection

Useful for detecting lines with width = 1.


Edge Detection

Points and lines are special cases of edges.

Edge detection is difficult since it is not clear what amounts to an edge!


Edge Detection


Impact of Noise


First & Second Derivatives of Edges

gradient, ofdirection :tan),(

gradient of magnitude :

/

/

1

22

x

y

yxyx

y

x

G

Gyx

GGGG

yf

xf

G

G

f

f


Edge Detection Operators

Figure 10.8, 10.9


Approximate Gradient with L1 Norm


Effects of Smoothing


Emphasizing Diagonal Edges

Use diagonal Sobel operator shown in figure 10.9(d)


Laplacian and Mexican Hat

2222 2/4

222/22

2

2

2

22

)(

rr e

rerh

y

f

x

ff

LoG operator


Comparison of Edge Detection

original Sobel LoG Threshold LoG Zero-crossing

Gaussian smooth operator

Laplacian operator

Gradient method: suitable for abrupt gray level transition, sensitive to noise

2nd order derivative: good for smooth edges


Boundary Extraction

Edge detection classifies individual pixels to be on an edge or not.

Isolated edge pixels is more likely to be noise rather than a true edge.

Adjacent or connected edge pixels should be linked together to form boundary of regions that segment the image.

Edge linking methods:» Local processing » Hough transform» Graphic theoretic method» Dynamic programming


Local Processing Edge Linking

An edge pixel will be linked to another edge pixel within its own neighborhood if they meet two criteria:

( , ) ( , )

( , ) ( , )

o o

o o

f x y f x y E

x y x y A


Global Processing Edge Linking: Hough Transform

Find a subset of n points on an image that lie on the same straight line.Write each line formed by a pair of these points as

yi = axi + bThen plot them on the parameter space (a, b):

b = xi a + yi

All points (xi, yi) on the same line will pass the same parameter space point (a, b).Quantize the parameter space and tally # of times each points fall into the same accumulator cell. The cell count = # of points in the same line.


Hough Transform in () plane

To avoid infinity slope, use polar coordinate to represent a line.

Q points on the same straight line gives Q sinusoidal curves in () plane intersecting at the same (ii) cell.

sincos yx


Example


Threshold Segmentation


Effect of Illumination on Thresholding


Threshold Example


Needs of Adaptive Threshold


Needs of Local Threshold

Properly and improperly segmented subimages from Fig. 10.30.

Further division of the sub-image, and result of adaptive thresholding


Threshold: Hypothesis Testing

Question: » Does this pixel with

intensity z belong to a region (edge) or not?

Hypothesis» H0: Null. It does not» H1: Alt. It does

Likelihood» p(z|zH0) = p1(z)» p(z| z H1) = p2(z)

Prior» P1 = p(zH0) , » P2 = p(zH1)

Maximum likelihood: » Pixel z belongs to a region if

p(z|H1) > p(z|H0)

Bayesian:

P2 p(z|H1) > P1 p(z|H0)

Sufficient statistic:z > T


Uni-model Gaussian Example

Given

Set

P1 p1(T) = P2 p2(T)

and solve for T. Take log on both sides

and simplify to

AT2 + BT + C = 0

2

2

2exp

2

1)(

i

ii

zzp

1

2

21

221

21

21

1222

21

21

22

22

21

212

221

22

21

ln2

then, If

ln2

2

P

PT

P

PC

B

A


Clustering Problem Statement

Given a set of vectors {xk; 1 k K}, find a set

of M clustering centers {w(i); 1 i c} such that each xk is assigned to a cluster, say, w(i*),

according to a distance (distortion, similarity) measure d(xk, w(i)) such that the average

distortion

is minimized. I(xk,i) = 1 if x is assigned to cluster i with cluster

center w(I); and = 0 otherwise -- indicator function.

c

i

K

kkk iWxdixI

KD

1 1

))(,(),(1


k-means Clustering Algorithm

Initialization: Initial cluster center w(i); 1 i c, D(–1)= 0, I(xk,i) = 0, 1 i c, 1 k K;

Repeat (A) Assign cluster membership (Expectation step)

Evaluate d(xk, w(i)); 1 i c, 1 k K

I(xk,i) = 1 if d(xk, w(i)) < d(xk, w(j)), j i; = 0; otherwise. 1 k K (B) Evaluate distortion D:

(C) Update code words according to new assignment (Maximization)

(D) Check for convergence if 1–D(Iter–1)/D(Iter) < , then convergent = TRUE,

ciixINxixIiWN

kki

N

kkk

1,),(,),()(11

KkiwxdixIiterDN

kkk

1))(,(),()(1


A Numerical Example

x = {1, 2,0,2,3,4},

W={2.1, 2.3}

Assign membership

2.1: {1, 2, 0, 2}

2.3: {3, 4}

Distortion

D = (1 2.1)2 + (22.1)2 + (0 2.1)2 + (2 2.1)2 + (3 2.3)2 + (4 2.3)2

3. Update W to minimize distortionW1 = (12+0+2)/4 = .25

W2 = (3+4)/2 = 3.5

4. Reassign membership.25: {1, 2, 0}3.5: {2, 3, 4}

5. Update W:

w1 = (1 2+0)/3 = 1w2 = (2+3+4)/3 = 3.

Converged.


Thresholding Example

Threshdemo.m

© 2002-2003 by yu hen hu 1 ece533 digital image processing image segmentation

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