Chapter 8 Image Representation & Analysis Chuan-Yu Chang ( 張傳育 )Ph.D. Dept. of Computer and Communication Engineering National Yunlin University of Science

Chapter 8Image Representation & Analysis

Chuan-Yu Chang ( 張傳育 )Ph.D.

Dept. of Computer and Communication Engineering

National Yunlin University of Science & Technology

[email protected]

http://mipl.yuntech.edu.tw

Office: EB212

Tel: 05-5342601 Ext. 4337

2醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.

Image Representation

To perform a computerized analysis of an image, it is important to establish a hierarchical framework of processing steps representing the image and knowledge domain.

The bottom-up analysis starts with the analysis at the pixels-level representation and moves up toward the understanding of the scene or the scenario.

The top-down analysis starts with the hypothesis for the presence of an object and then moves toward the pixel-level representation to verify or reject the hypothesis using the knowledge-based models.



Bottom-Up

Scenario

Scene-1 Scene-I

Object-1 Object-J

S-Region-1 S-Region-K

Region-1 Region-L

Pixel (i,j)

Edge-MEdge-1

Pixel (k,l)

Top-Down



Knowledge-based models can be used at different stages of processing. The knowledge of physical constraints and tissue

properties can be very useful in imaging and image reconstruction.

Anatomical locations of various organs in the body often impose a challenge in imaging the desired tissue or part of the organ.

An object representation model usually provides the knowledge about the shape or other characteristic features of a single objects for the classification analysis.


Image Reconstruction

ImageSegmentation

(Edge and Region)

Feature Extractionand

Representation

Classificationand

Object Identification

Analysisof Classified Objects

Multi-Modality/Multi-Subject/Multi-Dimensional

Registration, Visualization and Analysis

Raw Data from Imaging System

Single ImageUnderstanding

Multi-Modality/ Multi-Subject/Multi-Dimensional

Image Understanding

Scene Representation

Models

Object Representation

Models

Feature Representation

Models

Edge/Region Representation

Models

Physical Property/Constraint

Models

Knowledge Domain

DataDomain


Feature Extraction

After segmentation, specific features representing the characteristics and properties of the segmented regions in the image need to be computed for object classification and understanding.

There are four major categories of features for region representation: Statistical Features

Provide quantitative information about the pixels within a segmented region.

Ex: Histogram, Moments, Energy, Entropy, Contrast, Edges


Image Analysis: Feature Extraction Shape Features

Provide information about the characteristic shape of the region boundary.

Ex: Boundary encoding, Moments, Hough Transform, Region Representation, Morphological Features

Texture Features Provide information about the local texture within the region or

the corresponding part of the image. Ex: second-order histogram statistics, co-occurrence matrix,

wavelet processing. Relational Features

Provide information about the relational and hierarchical structure of the regions associated with a single or a group of objects.


Statistical Pixel-Level Features The histogram of the gray values of pixels

Mean of the gray values of the pixels

Variance and central moments in the region

where n=2 is the variance of the region.n=3 is a measure of noncentralityn=4 is a measure of flatness of the histogram.

n

rnrp i

i

1

0

1 L

iii rpr

nm

1

0

L

i

niin mrrp


Statistical Pixel-Level Features

Energy: Total energy of the gray-values of pixels

Entropy 熵

Local contrast

Maximum and minimum gray values

1

02log

L

iii rrpEnt

yxPyxP

yxPyxPyxC

sc

sc

,,,max

,,,

1

0

2L

iirpE


Shape Features

The shape of a region is defined by the spatial distribution of boundary pixels. Circularity, compactness, and elongatedness through the m

inimum bounded rectangle that covers the region. Several features using the boundary pixels of the segmente

d region can be computed as Chain code for boundary contour Fourier descriptor of boundary contour Central moments based shape features for segmented region Morphological shape descriptors


Some Shape Features

A

EH

D

B

C

FG

O

•Longest axis GE.•Shortest axis HF.•Perimeter and area of the minimum bounded rectangle ABCD.•Elongation ratio: GE/HF•Perimeter p and area A of the segmented region.

•Circularity

•Compactness

2

4

p

AC

A

pC p

2


Boundary Encoding :Chain Code

Define a neighborhood matrix with the orientation primitives with respect to the center pixel.

The code of specific orientation are set for 8-connected neighborhood directions.

The orientation directions are codes with a numerical value ranging from 0 to 7.

The boundary contour needs to be approximated as a list of segments that have pre-selected length and directions.



To obtain boundary segments representing a piecewise approximation of the original boundary contour, the “divide and conquer ” is applied. Selects two points on a boundary contour as vertices. A straight line joining the two selected vertices can be

used to approximate the respective curve segment if it satisfies a “maximum-deviation” criterion for no further division of the curve segment. The maximum deviation criterion is based on the

perpendicular distance between any point on the original curve segment between the selected vertices and corresponding approximated straight-line segment.



If the perpendicular distance or deviation of any point on the curve segment from the approximated straight-line segment exceeds a pre-selected deviation threashold, the curve segment is further divided at the point of maximum deviation.

This process of dividing the segments with additional vertices continues until all approximated straight-line segments satisfy the maximum-deviation criterion.

The representation is further approximated using the orientation primitive of the 8-connected neighborhood.

Two parameters can change the chain code: number of orientation primitives and the maximum deviation threshold used in approximating the curve.



The 8-connected neighborhood codes (left) and the orientation directions (right) with respect to the center pixel xc.

04

23 1

5 6 7

04

23 1

5 6 7

xc


A schematic example of developing chain code for a region with boundary contour ABCDE. From top left to bottom right: the original boundary contour, two points A and C with maximum vertical distance parameter BF, two segments AB and BC approximating the contour ABC, five segments approximating the entire contour ABCDE, contour approximation represented in terms of orientation primitives, and the respective chain code of the boundary contour.

FA D

C

E

B

A D

C

E

B

A D

C

E

B

A D

C

E

B

A

B C

DChain Code: 110000554455533

選取在方向及梯度上有較明顯的兩的頂點為起始點

BF 大於預設值，需將 AC 分成 AB, BC


Boundary Encoding: Fourier Descriptor Fourier series may be used to approximate a

closed boundary of a region. Assume that the boundary of an object is express

ed as a sequence of N points with the coordinates u[n]={x(n), y(n)}, such that

The discrete Fourier Transform of the sequence u[n] is the Fourier descriptor Fd[n] of the boundary and is defined as

niynxnu 1,,2,1,0 Nn

1

0

/21 N

n

Nnid enu

NnF 10 Nnfor


Boundary Encoding: Fourier Descriptor

Rigid geometric transformation of a boundary such as translation, rotation and scaling can be represented by simple operations on its Fourier transform.

The Fourier descriptors can be used as shape descriptors for region matching dealing with translation, rotation and scaling.


Moments for Shape Description The shape of a boundary or contour can be represented

quantitatively by the central moments for matching. The central moments represent specific geometrical pro

perties of the shape and are invariant to the translation, rotation and scaling.

The central moments pqof a segmented region or binary image f(x,y) are given by

L

i

L

j

qj

pipq yxfyyxx

1 1

,

L

i

L

jjii yxfxx

1 1

,

L

i

L

jjij yxfyy

1 1

,


Moments for Shape Description The normalized central moments are defined as

There are seven invariant moments for shape matching

12

qp

2

03212

123003210312

20321

21230123003217

03211230112

03212

123002206

20321

2123003210321

20321

21230123012205

20321

212304

20321

202203

211

202202

02201

33

33

4

33

33

3

4

00

pqpq


Morphological Processing for Shape Description

Mathematical morphology A tool for extracting image components that are use

ful in the representation and description of region shape, such as boundaries, skeletons, and convex hull.

Sets in mathematical morphology represent objects in an image.

2D integer space Z2

(x,y) coordinates Z3: gray-scale digital images

(x,y) coordinates, and gray-level value


Morphological Processing for Shape Description (cont.) Let A be a set in Z2, If a=(a1, a2) is an element of A

If a is not an element of A, we write

The set with no elements is called the null or empty set and denoted by the symbol .

The elements of the sets with which we are concerned are the coordinates of pixels representing objects. Ex:

set C is the set of elements, w, such that w is formed by multiplying each of the two coordinates of all the elements of set D by -1.

Aa

Aa

D} -d, for d { w | w C

(9.1-1)

(9.1-2)


Basic Concepts from Set Theory Subset

If every element of a set A is also an element of another set B, then A is said to be a subset of B.

Union The set of all elements belonging to either A, B, or both

Intersection The set of all elements belonging to both A and B

Morphological Processing for Shape Description (cont.)

BA

BAC

BAD

(9.1-3)

(9.1-4)

(9.1-5)



Disjoint (mutually excusive) If the two set have no common elements

Complement: The complement of a set A is the set of elements not

contained in A

Difference: the set of elements that belong to A, but not to B.

cBABwAwwBA },|{

BA

}|{ AwwAc

(9.1-6)

(9.1-7)

(9.1-8)




Preliminaries (cont.)

Reflection

Translation

},|{ˆ BbforbwwB

},|{)( AaforzaccA z

(9.1-9)

(9.1-10)


The principal logic operations used in image processing are AND, OR, and NOT

The three basic logical operations Performed on a pixel by pixel basis between corresponding

pixels of two or more images. Logical operation are restricted to binary variables

These operations are functionally complete in the sense that they can be combined to form any other logic operation

Logic Operations Involving Binary Images


–Black indicates a binary 1 –White indicates a 0.

Logic Operations Involving Binary Images


Dilation and Erosion For sets A and B in Z2

The dilation of A by B, denoted

where set B is referred to as the structuring element.

The dilation of A by B is the set of all displacements, z, such that and A overlap by at least one element.

}])ˆ[(|{

})ˆ(|{

AABz

ABzBA

z

z

B̂

(9.2-1)

(9.2-2)


Dilation and Erosion (cont.)



Set A

Set B

A large region with square shape representing the set A and a small region with rectangular shape representing the structuring element set B.


The dilation of set A by the structuring element set B (top left), the erosion of set A by the structuring element set B (top right) and the result of two successive erosions of set A by the structuring element set B (bottom).

: Dilation of A by B : Erosion of A by B

A A


Dilation and Erosion (cont.) Example of dilation

bridging gaps The maximum length of the breaks is known to be two pi

xels. A simple structuring element that can be used for repairi

ng the gaps is shown in Fig. 9.5(b)


Dilation and Erosion

For sets A and B in Z2

The erosion of A by B, denoted

where set B is referred to as the structuring element. The erosion of A by B is the set of all points z such

that B, translated by z, is contained in A.

})(|{ ABzBA z

BA


Dilation and Erosion


Morphological Features

A

B

BA

BA


Example of erosion-eliminating irrelevant detail

Dilation and Erosion 使用 13x13 的方形結構，對圖 (a) 進行 erosion

使用 13x13 的方形結構，對圖 (b) 進行 dilation


Opening and Closing Opening

Generally smoothes the contour of an object, breaks narrow isthmuses, and eliminates thin protrusions.

The opening A by B is the erosion of A by B, followed by a dilation of the result by B. View the structuring element B as a flat “rolling ball” The boundary of is then established by the points in B that reach

the farthest into the boundary of A as B is rolled around the inside of this boundary.

BA

ABB

BBABA

zz |

)(


Closing Tends to smooth sections of contours, fuses narrow breaks

and long thin gulfs, eliminates small holes, and fills gaps in the contour.

The closing of set A by structuring element B, denoted

The closing of A by B is simply the dilation of A by B, followed by the erosion of the result by B.

BBABA )(

Opening and Closing

BA


Opening and Closing


Opening and Closing The opening operation satisfies the

following properties AB is a subset of A If C is a subset of D,

then C B is a subset of D °B (A B) B=A B


Opening and Closing The properties of closing operation

A is a subset of AB If C is a subset of D,

then C B is a subset of D B (A B) B=A B

Multiple openings or closings of a set have no effect after the operator has been applied once.


Opening and Closing



The morphological opening and closing of set A (top left) by the structuring element set B (top right): opening of A by B (bottom left) and closing of A by B (bottom right).

A

B



Example of morphological operations on MR


Texture Features Texture is an important spatial property . There are three major approaches to represent texture

Statistical Based on region histograms, their extensions and their moments. Representing the high-order distribution of gray values in the image a

re used for texture representing. Structural

Arrangements of pre-specified primitives in texture representation, such as a repetitive arrangement of square and triangular shapes.

Spectral Based on the autocorrelation function of a region or on the power dis

tribution in Fourier transform domain. Texture is represented by a group of specific spatio-frequency compo

nents, such as Fourier and wavelet transform.


Texture Features Gray-level co-occurrence matrix (GLCM)

Exploits the high-order distribution of gray values of pixels that are defined with a specific distance or neighborhood criterion.

GLCM P(i,j) is the distribution of the number of occurrences of a pair of gray values i and j separated by a distance vector d=[dx, dy]

The GLCM can be normalized by dividing each value in the matrix by the total number of occurences providing the probability of occurrence of a pair of gray values separated by a distance vector.

Statistical texture features are computed from the normalized GLCM. The second-order histogram H(yq, yr, d) representing the pro

bability of occurrence of a pair of gray values yq and yr separated by a distance vector d.


Gray Level Co-occurrence Matrix (GLCM)

0o

45o

90o

135o

The four direction for the GLCM

1 1 2 2

1 1 2 2

3 3 1 1

3 3 1 1

GrayLevel 1 2 3

1 2 2 0

2 0 1 0

3 2 1 1Co-occurrence matrix for 45o


Gray Level Co-occurrence matrix (GLCM)

Figure 8.11. (a) A matrix representation of a 5x5 pixel image with three gray values; (b) the GLCM P(i,j) for d=[1,1].


Texture Feature Entropy of H(yq, yr, d)

The entropy is a measure of texture nonuniformity

Angular Second Moment of H(yq, yr, d) ASMH indicates the degree of homogeneity among textures

Contrast of H(yq, yr, d) (yq, yr) is a measure of intensity similarity

t

tq

t

r

y

yy

y

yyrqrqH dyyHdyyHS

1

,,log,, 10

t

tq

t

tq

y

yy

y

yyrqH dyyHASM 2,,

t

q

t

q

y

yy

y

yyrqrq dyyHyyContrast

1 1

,,,


Texture Feature Inverse Difference Moment of H(yq, yr, d), IDMH

Provides a measure of the local homogeneity among texture

Correlation of H(yq, yr, d) The correlation attribute is large for similar elements of the secon

d-order histogram.

t

q

t

q

y

yy

y

yy rq

rdH yy

dyyHIDM

1 1,1

,,

t

q

t

qrq

y

yy

y

yyrqrrqq

yyH dyyHyyyyCor

1 1

,,1

t

tr

y

yyrqqm dyyHdyH ,,,

t

q

y

yyrqrm dyyHdyH

1

,,,


Texture Feature Mean of H(yq, yr, d), Hm

The mean characterizes the nature of the gray-level distribution

Deviation of Hm(yq, d), Hm

Indicates the amount of spread around the mean of the marginal

distribution.

Entropy of Hd(ys, d), SHd(ys,d)

t

q

y

yyqqHm dyHmy

1

,

dyHdyHyy qm

y

yy

y

yyrmrqHm

t

q

t

r

,,1 1

2

t

s

s

y

yysdsddyHd dyHdyHS

1

,log, 10,


Texture Feature

Angular Second Moment of Hd(ys, d), ASM Hd(ys, d)

Mean of Hd(ys, d), Hd(ys, d),

t

s

s

y

yysddyHd dyHASM

1

2, ,

t

s

s

y

yysdsdyHd dyHy

1

,,


Benign lesion of X-ray mammogram

malignant lesion of X-ray mammogram

GLCM of Fig. (a)

GLCM of Fig. (b)


Relational Features

Relational features Provide information about adjacencies, repetitive patterns

and geometrical relationships among regions of an object. Could be extended to describe the geometrical

relationships among objects in an image or a scene. The relational features can be described in the form of

graphs or rules using a specific syntax or language The quad-tree based region descriptors can be used for

object recognition and classification using the tree matching algorithms.


Relational Features

Figure 8.13: A block representation of an image with major quad partitions (top) and its quad-tree representation.


Relational Features

A

C

B

D

F

I

E

B

C

A

I

ED

F

A tree structure representation of brain ventriclesfor applications in brain image segmentation and analysis


Feature and Image Classification

Features selected for image representation are classified for object recognition and characterization

Feature Based Pattern Classifiers Statistical Pattern Recognition

Unsupervised Learning Supervised Learning

Syntactical Pattern Recognition Logical predicates

Rule-Based Classifiers Model-Based Classifiers Artificial Neural Networks



Statistical Pattern Recognition Unsupervised Learning

Cluster the data based on their separation in the feature space.

K-means and fuzzy clustering methods Supervised Learning

It uses labeled clusters of training samples in the feature space as models of classes.

Nearest neighbor classifier, which assigns a data point to the nearest class model in the feature space.


Nearest Neighbor ClassifierA d i s t a n c e m e a s u r e )( fjD i s d e f i n e d b y t h e E u c l i d e a n d i s t a n c e i n t h e f e a t u r e s p a c e a s

jjD uff )(

w h e r e CjN

jcfj

jj ,...2,1

1

fu

i s t h e m e a n o f t h e f e a t u r e v e c t o r s f o r t h e c l a s s jc a n d N j i s t h e t o t a l n u m b e r o f f e a t u r e

v e c t o r s i n t h e c l a s s jc .

T h e u n k n o w n f e a t u r e v e c t o r i s a s s i g n e d t o t h e c l a s s ic i f

)]([min)( 1 ff jC

ji DD


Statistical classification Method

A probabilistic approach can be applied to the task of classification to incorporate a priori knowledge to improve performance. Bayesian and maximum likelihood methods have

been widely used in object recognition and classification.

Bayesian


Statistical classification Method The probability of a feature vector f belonging to th

e class i (ci)is denoted by p(ci /f). The average risk of wrong classification for assign

ing the feature vector to the class cj is defined as

A Bayes classifier assigns an unknown feature vector to the class cj if

C

kkkjj cpZr

1

ff k

C

kkkjj cPcpZr

1

ff

ff ji rr

q

C

qqqjk

C

kkki cPcpZcPcpZ

11

ff



Rule-Based Systems Analyzes the feature vector using multiple sets or

rules that are designed to check specific conditions in the database of feature vectors to initiate an action.

The rules are composed of two parts Condition premises Actions

They are based on expert knowledge to infer the action if the conditions are satisfied.



A rule-based system has three sets of rules Supervisory or strategy rules

Guide the analysis process and provide the control actions such as starting and stopping analysis.

Focus of attention rules Bring specific features into analysis by accessing and

extracting the required information or features from the database

Knowledge rules Analyze the information with respect to the required

conditions and implement an action causing changes in the output database.


A schematic diagram of a rule-based system for image analysis

Figure 8.15. A schematic diagram of a rule-based system for image analysis.


Strategy RulesStrategy Rule SR1:

If NONE REGION is ACTIVE NONE REGION is ANALYZED Then ACTIVATE FOCUS in SPINAL_CORD AREA Strategy Rule SR2:

If ANALYZED REGION is in SPINAL_CORD AREA ALL REGIONS in SPINAL_CORD AREA are NOT ANALYZED Then ACTIVATE FOCUS in SPINAL_CORD AREA Strategy Rule SR3:

If ALL REGIONS in SPINAL_CORD AREA are ANALYZED ALL REGION in LEFT_LUNG AREA are NOT ANALYZED

Then ACTIVATE FOCUS in LEFT_LUNG AREA


FOA Rules

Focus of Attention Rule FR1:

If REGION-X is in FOCUS AREA REGION-X is LARGEST REGION-X is NOT ANALYZED

Then ACTIVATE REGION-X

Focus of Attention Rule FR2:

If REGION-X is in ACTIVE MODEL is NOT ACTIVE

Then ACTIVATE KNOWLEDGE_MERGE rules


Knowledge Rules

Knowledge Rule: Merge_Region_KR1 If

REGION-1 is SMALL REGION-1 has GIGH ADJACENCY with REGION-2 DIFFERENCE between AVERAGE VALUE of REGION-1 and

REGION-2 is LOW or VERY LOW REGION-2 is LARGE or VERY LARGE

Then MERGE REGION-1 in REGION-2 PUT_STATUS ANALYZED in REGION-1 and REGION-2


Image and Feature Classification: Neural Networks

Several neural networks have been used for feature classification for object recognition and image interpretation. Backpropagation Radial Basis Function Associative Memories Self-Organizing Map

d

tii wwx

10


Neuro-Fuzzy pattern Classification

The process of network training could be seen as the attempt at finding an optimal dichotomy of the input space into these convex regions.

The classes are separated in the feature space by computing the homogeneous non-overlapping closed convex subsets.

The classification is obtained by placing separating hyperplanes between neighboring subsets representing classes.

Grohman and Dhawan Fuzzy membership function Mf is dervised for each convex subse

t. The classification decision is made by the output layer based on t

he “winner-take-all” principle. The resulting category C is the convex set category with the high

est value of membership function for the input pattern.


Convex sets Convex sets

A convex set is a set of elements from a vector space such that all the points on the straight line between any two points of the set are also contained in the set.

A set S in n-dimensional space is called a convex set if the line segment joining any pair of points of S lies entirely in S. If the set does not contain all the line segments, it is called concave.


Convex sets Convex Hull

The convex hull of a set of points is the smallest convex set that includes the points. For a two dimensional finite set the convex hull is a convex polygon.

http://www.cse.unsw.edu.au/~lambert/java/3d/ConvexHull.html


Some basic Morphological Algorithm Convex Hull

A set A is said to be convex if the straight line segment jointing any two points in A lies entirely within A.

The convex hull H of an arbitrary set S is the smallest convex set containing S.

The set difference H-S is called the convex deficiency of S. The convex hull and convex deficiency are useful for object

description. Let Bi, i=1, 2, 3, 4, represent the four structuring elements in Fig.

9.19 (a). The procedure consists of implementing the equation:

where Let . Then the convex hull of A is

4

1

)(

i

iDAC

,...3,2,1 and 4,3,2,11 kiABXX ik

ik

AX i 0

iconv

i XD

(9.5-5)

(9.5-4)


Some basic Morphological Algorithm

The procedure consists of iteratively applying the hit-or-miss transform to A with B1; when no further changes occur, we perform the union with A and call the result D1.

The procedure is repeated with B2 until no further changes occur, and so on.

The union of the four resulting D’s constitutes the convex hull of A.

X indicates don’t care


Some basic Morphological Algorithm


Convex sets

Convex hull 演算法一 : Jarvis's March (gift wrapping) 找出最下方的點 p0. 它一定在 convex hull 的邊界上 .

找出 p1, 使 p0 與 p1 的連線與正 x 軸的夾角 ( 有向角 ) 最小 .

找出 p2, 使 p2 與 p1 的連線與正 x 軸的夾角最小 .

... 直到回到 p0 為止 .


Convex sets

Convex hull 演算法二 : Graham's scan 找出最下方的點 p0. 它一定在 convex hull 的邊界上 . 以「 p0 到各點的射線與 x 軸的夾角」作為比較的依據 ,

對所有的點排序 . 依序如下檢查 p1, p2,.... 檢查 pi 時要做的事情 : 看看 stack 上第二高的元素 , stac

k 上最頂端的元素 , 與 pi 三點兩射線是左轉還是右轉 . 如果是右轉 , 就 pop, 並重複此步驟 .

push pi. 檢查下一個 pi.


Neuro-Fuzzy pattern Classification The neuro-fuzzy pattern classifier design method incl

udes three stages Convex set creation Hyperplane placement

hyperplane layer creation Construction of the fuzzy

membership function foreach convex set. Generation of the fuzzy

membership function layer.

M1

winner-take-alloutput layer

L

1

fuzzy membershipfunction layer

x1

xi

xd

hyperplanelayer

inputlayer

max

M2

MK

C



There are two requirements for computing the convex sets Homogeneous

Need to devise a method of finding one-category points within another category’s hull. How to find whether the point P lies inside of a convex hull

(CH) of points. How to find out if two convex hulls of points are

overlapping.

Non-overlapping.


Neuro-Fuzzy pattern Classification Algorithm A1 addresses the first problem using the se

paration theorem, which states that for two closed non-overlapping convex sets S1 and S2 there always exists a hyperplane that separates the two sets. 1. Consider P as origin. 2. Normalize points of CH 3. Find min and max vector coordinates in each dimension. 4. Find set E of all vectors V that have at least one extreme c

oordinate. 5. Compute mean and use it as projection vector :

Evv ii


Neuro-Fuzzy pattern Classification 6. Set a maximum number of allowed iterations (usually=2

n) 7. Find a set U=(u1, u2,…, um) (where m<=n) of all points in C

H that have negative projection on . 8. If U is empty (P is outside of CH) exit, else proceed to St

ep 9. 9. Compute coefficient as:

10. Calculate correction vector by computing all of its k-dimensional components:

11. Update , where >1 is a training parameter. 12. If iteration limit exceed exit, otherwise go to step 7.

dU

dUU

kk

kkk

0

0

UT

m

iiu

mU

1

1



Algorithm A2: Convex subset creation 1. select one class category from the training set and consi

der all data points in the category. 2. Construct the convex subsets.

Add the current point P to the subset S. Loop over points from negative category. UpdateΛ.

3. If all points in the category have been assigned to a subset proceed to step 4, otherwise go back to Step 2 and create the next convex subset.

4. Check if all categories have been divided into convex subsets. If not, go back to Step 1 and create subsets of the next category.


Figure 8.18. The structure of the fuzzy membership function.


Figure 8.19. Convex set-based separation of two categories.


Figure 8.20. (a). Fuzzy membership function M1(x) for the subset #1 of the black category. (b). Fuzzy membership function M2(x) for the subset #2 of the black category.


Figure 8.21. Fuzzy membership function M3(x) (decision surface) for the white category membership.

Figure 8.22. Resulting decision surface Mblack(x) for the black category membership function


Image analysis example It is difficult to distinguish between benign and malig

nant microcalcifications associated with breast cancer.

Dhawn used the second-order histogram statistics and wavelet processing to represent texture for classification into benign and malignant. Two sets of ten wavelet features were computed for discret

e Daubechies filter prototypes. 40 features were extracted and used in a Genetic algorithm

based feature reduction and correlation analysis. 10 binary segmented microcalcification cluster features 10 global texture based image structure features. 20 wavelet analysis based local texture features. (see Page. 242~245)


Image analysis example Genetic algorithm were used to select the best subset of

features from the binary cluster, global and local texture representation.

GA is a robust optimization and search method based on natural selection principles.

GA generate a population of individuals through selection, and search for the fittest individuals through crossover and mutation.

They operate on a representation of problem parameters, rather than manipulating the parameters themselves.

These parameters are typically encoded as binary strings that are associated with a measure of goodness, or fitness value.


Image analysis example Through the process of reproduction, individual strings are

copied according to their degree of fitness. Once the parent population is selected through reproduction, the

offspring population is created after application of genetic operators.

The purpose of crossover is to discover new regions of the search space rather than relying on the same population of strings.

In crossover, strings are probabilistically mated by swapping all characters located after a randomly chosen bit position.

Mutation is a secondary genetic operator that randomly changes the value of a string position to introduce variation in the population and recover lost genetic information.

Mutation preserves the random nature of the search process and regenerates fit strings that may have been destroyed or lost during crossover or reproduction.

The mutation rate controls the probability that a bit value will be changed.


Image analysis example

Using the GA algorithm, the initial set of 40 features was reduced to the two best correlated set of 20 features.

The selected features were used as inputs to the radial basis function for subsequent classification of the microcalcification.

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Chapter 8 Image Representation & Analysis Chuan-Yu Chang ( 張傳育 )Ph.D. Dept. of Computer and Communication Engineering National Yunlin University of Science