Chapter 8 Image Representation & Analysis Chuan-Yu Chang ( 張傳育 )Ph.D. Dept. of Computer and...
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Transcript of Chapter 8 Image Representation & Analysis Chuan-Yu Chang ( 張傳育 )Ph.D. Dept. of Computer and...
Chapter 8Image Representation & Analysis
Chuan-Yu Chang ( 張傳育 )Ph.D.
Dept. of Computer and Communication Engineering
National Yunlin University of Science & Technology
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.
3醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Image Representation
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
4醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Image Representation
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.
5醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
6醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
7醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.
8醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
9醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
10醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
11醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
12醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.
13醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Boundary Encoding :Chain Code
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.
14醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Boundary Encoding :Chain Code
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.
15醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Boundary Encoding :Chain Code
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
16醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
17醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
18醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.
19醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
,
20醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
21醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
22醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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)
23醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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)
24醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Morphological Processing for Shape Description (cont.)
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)
25醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Morphological Processing for Shape Description (cont.)
26醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Preliminaries (cont.)
Reflection
Translation
},|{ˆ BbforbwwB
},|{)( AaforzaccA z
(9.1-9)
(9.1-10)
27醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
28醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
–Black indicates a binary 1 –White indicates a 0.
Logic Operations Involving Binary Images
29醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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)
30醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Dilation and Erosion (cont.)
31醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Morphological Processing for Shape Description
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.
32醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
33醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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)
34醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
35醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Dilation and Erosion
36醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Morphological Features
A
B
BA
BA
37醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
38醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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 |
)(
39醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
40醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Opening and Closing
41醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
42醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.
43醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Opening and Closing
44醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Morphological Processing for Shape Description
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
45醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
46醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Example of morphological operations on MR
47醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.
48醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.
49醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.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
50醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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].
51醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
,,,
52醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
,,,
53醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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,
54醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
,,
55醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Benign lesion of X-ray mammogram
malignant lesion of X-ray mammogram
GLCM of Fig. (a)
GLCM of Fig. (b)
56醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.
57醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Relational Features
Figure 8.13: A block representation of an image with major quad partitions (top) and its quad-tree representation.
58醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
59醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
60醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Feature and Image Classification
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.
61醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
62醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
63醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
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64醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Feature and Image Classification
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.
65醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Feature and Image Classification
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.
66醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.
67醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
68醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
69醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
70醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
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71醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.
72醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.
73醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
74醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
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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)
75醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
76醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Some basic Morphological Algorithm
77醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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 .
78醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.
79醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
80醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Neuro-Fuzzy pattern Classification
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.
81醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
82醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
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83醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Neuro-Fuzzy pattern Classification
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.
84醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Figure 8.18. The structure of the fuzzy membership function.
85醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
Figure 8.19. Convex set-based separation of two categories.
86醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.
87醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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
88醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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)
89醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.
90醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.
91醫學影像處理實驗室 (Medical Image Processing Lab.) Chuan-Yu Chang Ph.D.
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.