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Computer and Robot Vision I
Chapter3
Binary Machine Vision:
Region Analysis
Instructor: Shih-Shinh Huang
1
Contents
 Region Properties
 Simple Global Properties
 Extremal Points
 Spatial Moments
 Mixed Spatial Gray Level Moments
 Signature Properties
 Contour-Based Shape Representation
2
Computer and Robot Vision I
Introduction
Region Properties
3
Region Properties
Introduction
 Region Description
 Region is a segment produced by connected
component labeling or signature segmentation.
 The computation of region properties can be the
input for further classification.
• Gray-Level Value Analysis
• Shape Property Analysis
4
Region Properties
Simple Global Properties
 Region Area A
A
1
( r ,c )R
A=21
r=3.476
c=4.095
 Centroid ( r , c )
1
r 
r

A ( r ,c )R
0 1 2 3
0
1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
2
0 0 1 1 1 1 1 0
3
0 0 1 1 1
4
0 0 1 1 1
0 1 1 1 1 1 0 0
0 1 1 0 0 1 1 0
0 0 0 0 0 0 0 0
5
1
c 
c

A ( r ,c )R
6
7
5
4 5 6 7
0 0 0
0 0 0
Region Properties
Simple Global Properties
 Perimeter Description
 It is a sequence of its interior border pixels.
 Border pixels are the pixels that have some
neighboring pixel outside the region.
 Types of Perimeter
 4-Connected Perimeter P4: Use 8-Connectivity to
determine the border pixel.
 8-Connected Perimeter P8:Use 4-Connectivity to
determine the border pixel.
6
Region Properties
Simple Global Properties
 4-Connected Perimeter P4
P44 {(
{(r1,,c2)),
),8((2r, 2
2,3
P
(R2,|1N
c)),(R
),}(3,2)}
(2,1)  P4
7
(2,2)  P4
Region Properties
Simple Global Properties
 8-Connected Perimeter P8
P88 {({(r1
),R
(2| ,1
,2)}
P
, c,2
)
N),
, c,3) ),
 (R3
}
4 ((r2
(2,1)  P8
8
(2,2)  P8
Region Properties
Simple Global Properties
 Perimeter Representation  P 
 It is a sequences of border pixels in
 P  (r0 , c0 ), (r1, c1 ),....(rK 1, cK 1 )
•
(rk 1 , ck 1 ), (rk , ck )
•
(r0 , c0 )  (rK 1 , cK 1 )
 P4 
are neighborhood
 P8 
9
P4
or
P8
Region Properties
Simple Global Properties
 Perimeter Length | P |
Vertical or Horizontal Line
Diagonal Line
| P4 | 8
| P8 | 4 2
10
Region Properties
Simple Global Properties
 Compactness Measure
| P |2 / A
 It is used as a measure of a shape’s compactness.
 Its smallest value is not for the digital circularity,
but it would for continuous planar shapes
• Octagons
• Diamonds
11
Region Properties
Simple Global Properties
 Circularity Measure
R /  R
(r , c )
 Boundary Pixels
(r1 , c1 )
 P  (r0 , c0 ), (r1, c1 ),....(rK 1, cK 1 )
1
R 
K
1
 
K
2
R
(r0 , c0 )
K
 || (r , c
k 0
k
)  (r , c ) ||
k
K
 || (r , c
k 0
k
(r2 , c2 )
k
)  (r , c ) ||   R 
2
12
(r3 , c3 )
Region Properties
Simple Global Properties
 Circularity Measure
R /  R
 Properties
• Digital shape  circular,  R /  R increases monotonically.
• It is similar for similar digital/continuous shapes
• It is orientation and area independent.
 Polygon Side Estimation
 R
N  1.41111 
R



13
0.4724
Region Properties
Simple Global Properties
 Gray-Level Mean 
1

I (r , c)

A ( r ,c )A
 Gray-Level Variance 
1

1
2
2
2


 
I
(
r
,
c
)



I
(
r
,
c
)


 


A ( r ,c )A
A
 ( r ,c )A

2
Right hand equation lets us compute variance with
only one pass
14
Region Properties
Simple Global Properties
 Microtexture Properties
 Co-occurrence Matrix
P( g1 , g 2 ) 
P (.,.)
# [(ri , ci ), (rj , c j )] S | I (ri , ci )  g1 , I (rj , c j )  g 2 
#S
• S : a set of all pairs of pixels that are in some defined
spatial relationship (4-neighbors)
15
Region Properties
Simple Global Properties
0
0
1
2
3
1
2
3
P (.,.)
0
P1,0
P0,1
P2,2
DC & CV Lab.
16
Region Properties
Simple Global Properties
 Microtexture Properties
 Texture Second Moment
M 
M
2
P
 ( g1 , g2 )
g1 , g 2
 Texture Entropy
E
E    P( g1 , g 2 ) log P( g1 , g 2 )
g1 , g 2
 Texture Homogeneity H
P( g1 , g 2 )
H  
g1 , g 2 k  | g1 17g 2 |
Region Properties
Simple Global Properties
 Microtexture Properties
 Contrast
C
C
2
|
g

g
|
P
( g1 , g 2 )
 1 2
g1 , g 2
 Correlation


2
(
g


)(
g


)
P
(
g
,
g
)
/

 1
2
1
2
g1 , g 2
18
Region Properties
Extremal Points
 Definition of Extremal Points
 It has an extremal coordinate value in either its row
or column coordinate position
 They can be as many as eight distinct extermal points.
19
Region Properties
Extremal Points
20
Region Properties
Extremal Points
Different extremal points may be
coincident
21
Region Properties
Extremal Points
 Definition of Extremal Coordinate
Topmost
r1  r2  rmin  min{r | (r, c)  R}
Bottommost
r5  r6  rmax  max{r | (r, c)  R}
Leftmost
c7  c8  cmin  min{c | (r, c)  R}
Rightmost
c3  c4  cmax  max{c | (r, c)  R}
Topmost
Rightmost
Leftmost
Bottommost
22
Region Properties
Extremal Points
 Definition of Extremal Coordinate
Topmost Left
(r1 , c1 )  {rmin , min{c | (rmin , c)  R}
Topmost Right
(r2 , c2 )  {rmin , max{c | (rmin , c)  R}
Topmost Right
Topmost Left
23
Region Properties
Extremal Points
 Respective Axes (M1, M2, M3, M4)
 Form by each pair of opposite extremal points
• M1: Topmost Left
&
Bottommost Right
• M2: Topmost Right
&
Bottommost Left
• M3: Rightmost Top
&
Leftmost Bottom
• M4: Rightmost Bottom&
Leftmost Top.
 Properties
• Length
• Orientation
24
Region Properties
Extremal Points
25
Region Properties
Extremal Points
 Length of Respective Axes
M  (ri  rj ) 2  (ci  c j ) 2  Q( )
Quantization Error
Compensation Term
 (ri , ci ) : one end point of respective axes
 (rj , c j ): the other point of respective axes
26
Region Properties
Extremal Points
 Orientation of Respective Axes

 Orientation of a line segment is taken as
counterclockwise with respect to column axis.
  t an
1
(ri  rj )
 (ci  c j )
Quantization Error
Compensation Term
 1

| cos |
Q( )  
1


 | sin  |
27
|  | 45
|  | 45
Region Properties
Extremal Points
 Properties of Line-like Region
*
M
 Major Axis
: the axis with the largest length.
M *  max{M1, M 2 , M3 , M 4}
 The length and orientation of major axis stands
for the same thing for this region.
28
Region Properties
Extremal Points
• Properties of Line-like Region
M1, M 2, M3
M4
29
Region Properties
Extremal Points
 Properties of Triangular Shapes
 Apex Selection: Find the extremal point having
the greatest sum of its two largest distances.
• Extremal Point Distance
M ij  (ri  rj ) 2  (ci  c j ) 2
• Objective Function
(k1* , k2* , k3* )  argmax{M k1k2  M k1k3 }
30
Region Properties
Extremal Points
 Properties of Triangular Shapes
 Side Length L
L  (M k1k2  M k1k 3 ) / 2
L
h
 Base Length B
B  M k2k3
 Altitude Height
h
B
h
L2  B / 2 
2
31
Region Properties
Extremal Points
h
B
32
Region Properties
Spatial Moments
 Second-Order Spatial Moments
 Row Moment
 rr 
 rr
1
(r  r ) 2

A ( r ,c )R
 Mixed Moment
 rc 
 rc
1
(r  r )(c  c )

A ( r ,c )R
 Column Moment  cc
cc
1
2

(
c

c
)

A ( r ,c )R
33
Region Properties
Spatial Moments
 Second-Order Spatial Moments
 They have value meaning for a region of any shape
 Similarly, the covariance matrix has value and
meaning for any two-dimensional pdf.
 Example: An ellipse A whose center is the origin.
R  {(r , c) | dr2  2erc  fc 2  1}
d

e

e
  cc
1



2

f 
4(  rr  cc   rc ) 
   rc
34
  rc 

 rr 

Region Properties
Mixed Spatial Gray Level Moments
 Description
 A property that mixes up two properties.
• Spatial Properties: Region Shape, Position
• Intensity properties
 Two Second-order Mixed Spatial Gray Properties
 rg
1

(r  r )(I (r , c)   )

A ( r ,c )R
cg 
1
(c  c )(I (r , c)   )

A ( r ,c )R
35
Region Properties
Mixed Spatial Gray Level Moments
 Application: Determine the least-square, best-fit
gray level intensity plane.
I (r, c)   (r  r )   (c  c )  
 Unknowns Variables:
,  ,
 Objective Function
(ˆ , ˆ , ˆ )  arg min  2
( ,  , )
[ (r  r )   (c  c )    I (r, c)]2
36
Region Properties
Mixed Spatial Gray Level Moments
 Application: Determine the least-square, best-fit
gray level intensity plane
 Take partial derivative of
 2 with respect to ( ,  ,  )
(ˆ , ˆ , ˆ )  arg min  2
( ,  , )


(r  r ) 2

 ( r ,c )R

(r  r )(c  c )
 ( r
,c )R

(r  r )


 ( r ,c )R





(r  r )(c  c )  (r  r )
(r  r )(I (r , c)) 


    ( r ,c )R

( r ,c )R
( r ,c )R
2
(
c

c
)
(c  c )       (c  c )(I (r , c)) 


   ( r ,c )R

( r ,c )R
( r ,c )R
   

(
c

c
)
1
I
(
r
,
c
)






( r ,c )R
( r ,c )R
(
r
,
c
)

R



Least Square
Method
37
Region Properties
Mixed Spatial Gray Level Moments
 Application: Determine the least-square, best-fit
gray level intensity plane

 



2
2
  (r





 (rr)  r )  (r
 (rr)(c r)(cc)  c ) (0r  r )
(r
 r()rI(rr,)(
c)I (r , c)) 

    ( r ,c )R( r ,c )R
 (r ,c )R( r ,c )R


( r ,c )R( r ,c )R
( r ,c )R





2
2
  (r  (rr)(c r)(cc)  c )
(
c

(
c
c
)

c
)
(
0
c
 c)  
 (c
 c(c) I(rc,)(
c)I(r , c)) 







 ( r ,c )R( r ,c )R
  ( r ,c )R( r ,c )R


( r ,c )R( r ,c )R
( r ,c )R




  (r  r0)

      I (r , c)I (r , c)

0
A
(
c

c
)
1






 
 



(
r
,
c
)

R
(
r
,
c
)

R
(
r
,
c
)

R
(
r
,
c
)

R
(
r
,
c
)

R

 



 (r  r )  0
( r ,c )R
 (c  c )  0
 
1
I (r , c)  

A ( r ,c )R
( r ,c )R
38
Region Properties
Mixed Spatial Gray Level Moments


2




(
r

r
)
(
r

r
)(
c

c
)
0
(
r

r
)
I
(
r
,
c
)




0


 (rr







   ( r ,c )R
rc
rg

r ,c )R
( r ,c )R










2


(
r

r
)(
c

c
)
(
c

c
)
0


(
c

c
)
I
(
r
,
c
)




0






 rc
( r ,c )R  cg 
cc
 ( r ,c )R

( r ,c )R



  0   A   

 0
A
0 0
I
(
r
,
c
)
  





(
r
,c )R


 rg
 cg
 
 rr
 rc
 rc
 cc
 rc
 cc
 rr
 rc
 
 rr
 rc
39
 rg
 cg
 rc
 cc
Computer and Robot Vision I
Introduction
Signature Properties
40
Signature Properties
Introduction
 Signature Review
Remark: Signature analysis is important
because of easy, fast implementation in pipeline hardware
41
Signature Properties
Signature Computation
 Centroid ( r , c )
r
1
1
1
1
r

r

r
1

rPH (r )






A ( r ,c )R
A r {c|( r ,c )R} A r {c|( r ,c )R} A r
c
1
1
1
1
c

c

c
1

cPV (c)






A ( r ,c )R
A c {r|( r ,c )R} A r {r|( r ,c )R} A r
 Second-Order Moment (  rr )
 rr 

1
1
2
(
r

r
)

(r  r ) 2 



A ( r ,c )R
A r {c|( r ,c )R}
1
1
2
(
r

r
)
1

(r  r ) 2 PH (r )



A r
A r
{c|( r ,c )R}
42
Signature Properties
Signature Computation
 Second-Order Moment ( cc )
cc 

1
1
2
2
(
c

c
)

(
c

c
)




A ( r ,c )R
A r {c|( r ,c )R}
1
1
2
2
(
c

c
)
1

(
c

c
)
PV (c)



A r
A r
{c|( r ,c )R}
43
Signature Properties
Circle Center Determination
 Description
 We can determine the center position of circular
region from signature analysis.
cos y
cos x
| x |
y
x
| y |
44
A B C  D

A B C  D

cos x
cos y
Signature Properties
Circle Center Determination
 Derivation
A  B  r  r 
2
A B

A
1 2
1
r ( 2 )  rd sin   2
2
2
A
1 2
d

r 2  2 sin  
2 
r

A
1 2
r 2  2 cos  sin  
2
A
1 2
r 2  sin 2 
2
45
Signature Properties
Circle Center Determination
 Derivation
r
A
A B

1 2
r 2  sin 2 
2
2A
 2  sin 2
A B
Compute by a table-look-up technique
d  r cos 
A B

cos
46
Circle Center Determination
 Algorithm
 Step 1: Partition the circuit into four
quadrants formed by two orthogonal
lines intersecting inside the circle.
y
x
 Step 2: Using signature analysis to
compute the areas A, B, C, and, D.
 Step 3: Compute x, y using the
derived equation.
| x |
47
A B C  D

cos x
Computer and Robot Vision I
Introduction
Contour-Based
Shape Representation
48
Chain Code
 Description
 It describes an object by a sequence of unit-size
line segment with a given orientation.
 The first element must bear information about its
position to permit region reconstruction.
Chain Code: 3, 0, 0, 3, 0, 1, 1, 2, 1, 2, 3, 2
49
Chain Code
 Matching Requirement
 It must be independent of the choice of the first
border pixel in the sequence.
 It requires the normalization of chain code
• Interpret the chain code as a base 4 number.
• Find the pixel in the border sequence which results in the
minimum integer number.
Chain Code: (300301121232)4
Chain Code: (003011212323)4
50
Curvature
 Description
 Curvature is defined as the rate of change of slope
in the continuous case.
 The evaluation algorithm in the discrete case is
based on the detection of angles between two lines.
 Values of the curvature at all boundary pixels can
be represented by a histogram for matching.
51
Curvature
b: sensitivity to local changes.
52
Signature
 Description
 The signature is a sequence of normal contour
distances.
 It can be calculated for each boundary elements as
a function of the path length.
53
Chord Distribution
 Description
 Chord is a line joining any two points of the region
boundary.
 The distribution of lengths and angles of all chords
may be used for shape description.
 Definition of Chord Distribution
 b( x, y)  1 : contour points
 b( x, y )  0 : all other points
h(x, y)   b(i, j )b(i  x, j  y)
i
j
54
Chord Distribution
h(x, y)   b(i, j )b(i  x, j  y)
i
j
Rotation-Independent Radial Distribution
 /2
hr ( r ) 
 h(x, y )rd

/2
55
r  x 2  y 2
  sin 1 (y / r )
Segment Sequence
 Description
 It is a way to represent the boundary using
segments with specified properties.
 Recursive Boundary Splitting
56
Segment Sequence
 Structure Description
 Curves are segmented into several types
• Circular Arcs
• Straight Line
 Segments are considered as primitives for
syntactic shape recognition
Chromosomes Representation.
57
Scale-Space Image
 Description
 Sensitivity of shape descriptors to scale (image
resolution) is an undesirable feature.
 Some curve segmentation points exist in one
resolution and disappear in others.
 Approach Properties
 Only new segmentation points can appear at
higher resolution.
 No existing segmentation points can disappear.
58
Scale-Space Image
 Approach Description
 It is based on application of a unique Gaussian
smoothing kernel to a one-dimensional signal.
 The zero-crossing of the second derivative is
detected to determine the peak of curvature.
 The positions of zero-crossing give the positions of
curve segmentation points.
59
Scale-Space Image
60
Computer and Robot Vision I
The End
61