數位影像中熵的計算與應用
義守大學
資訊工程學系
黃健興
Outline

Entropy

Applications

Conclusions
• Definition
• Entropy of images
• Visual Surveillance System
• Background Extraction
Concept of Entropy


Rudolf Julius Emanuel Clausius , 1864
化學及熱力學
• 測量在動力學方面不能做功的能量總數
• 計算一個系統中的失序現象
• 描述系統狀態的函數
• 經常用熵的參考值和變化量進行分析比較
Information Theory


Claude Elwood Shannon , 1948
運用機率論與數理統計的方法研究資訊
•
•
•
•
•
•
•
編碼學
密碼學與密碼分析學
數據傳輸
數據壓縮
檢測理論
估計理論
數據加密
Definition


H ( X )  E ( I ( X ))
• E is the expected value,
• I is the information content of X.
n
n
i 1
i 1
H ( X )   p( xi ) I ( xi )   p( xi ) log b p( xi )
• p denotes the probability mass function of X
Advantage

Whole Image

Histogram

Entropy
• M×N Matrix
• N×1 Vector
• Single value
Entropy of Image

Pixel Color
Pixel Distribution

Texture

• Horizontal
• Vertical
The Statistic of gray-level
Position Information




Normalize the size of image
Edge Detection
• Sobel
• Canny
Horizontal Projection
Vertical Projection
Sobel Edge Detection

Sobel Filter
m1
m2
m3
m4
m5
m6
m7
m8
m9
 1  2
0
0

 1
2
 1
0 
1 
 1
 2


 1
0
 2
 1


 0
1
0
1
45
0
0
0
1
2

1

90
0
1

2

 0
 1

 2
1
0
1
2
1 
0 
 135
Sobel Edge Detection(cont.)
s0  m7  2m8  m9   m1  2m2  m3 
s45  m6  2m8  m9   m1  2m2  m4 
s90  m3  2m6  m9   m1  2m4  m7 
s135  m2  2m3  m6   m4  2m7  m8 
S  s0  s45  s90  s135  T
S  s0  s45  s90  s135  T
Sobel Edge Detection(cont.)
Horizontal Projection
0
0
0

0

0
0

0
0

0


 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
0
1
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0








0
0
0
0
0
0
0
0
0  0
0  0 
0  0

0  0
0  0

0  0
0  0

0  0
  0

0  0 
240
0
0 7 2 2 2 7 0  0
Horizontal Projection(cont.)
Vertical Projection
0
0

0

0
0

0
0

0


 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
0
1
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0








0
0
0
0
0
0
0
0
0  0
0  0 
0  0

0  0
0  0

0  0
0  0

0  0
  0

0  0 
0 7 2 2 2 2 2 7 0  0
0
320
Vertical Projection(cont.)
Local Binary Pattern

Pattern Texture T  t ( g 0  g c ,..., g p1  g c )
• Pattern T  t ( g , g ,..., g )
• Center Pixel gc
• Surrounding Pixel gi(i=0, 1,…,p-1)
• T  t (s( g  g ),..., s( g  g ))
c
0
c
p 1
0
p 1
c
1 x  0
s ( x)  
0 x  0
• Label
p 1
LBPP ,R ( xc , yc )   s( g p  g c )2 p
p 0
Local Binary Pattern(cont.)
g0
g1
g2
0
0
0
g3
gc
g4
1 gc
1
g5
g6
g7
0
0
0
g0  gc  0
g1  g c  0
g2  gc  0
g3  gc  0
gc
g4  gc  0
g5  gc  0 g6  gc  0 g7  gc  0
LBPP, R ( xc , yc )  0  20  0  21  0  22  1 23  1 24  0  25  0  26  0  27  24
Definition


H ( X )  E ( I ( X ))
• E is the expected value,
• I is the information content of X.
n
n
i 1
i 1
H ( X )   p( xi ) I ( xi )   p( xi ) log b p( xi )
• p denotes the probability mass function of X
Applications

Visual Surveillance System

Background Extraction
• variance of video information
• Block for pixel
Visual Surveillance System
F2
F 20
F 63
F 45
F 68
F 60
F 69
Visual Surveillance System
Gray Prediction – GM(1,1)
Gray Prediction – GM(1,1) (cont.)


Step 1: X 0   x0  1, x0  2,, x0  n
1
1
1
1




n

X

x
1
,
x
2
,

,
x
Step 2:
k
x 1 k    x 0  i 
i 0
k  1,2,  , n


1 1
1




k  1
z
k

x
k

x
 Step 3:
2
k  2,3,, n
1
Gray Prediction – GM(1,1) (cont.)


Step 4:
Step 5:
  z 1 2 
 1
 z 3

B
 
 1
 z n 
 x 0  2 
 0  
x 3

Y
  
 0  
 x n 
1

1


1
x 0  k   az 1 k   u

a
a
u

 B B
T

1
BT YN
Gray Prediction – GM(1,1) (cont.)

Step 6:
dx t 
 ax t   u
dt
 1
u   ak u
 0 




k

1

x
1


e 
x
a
a

k  1,2,, n

 0 
 1
 1
Step 7: x k  1  x k  1  x k 
 0 


u
 0 
a
 ak




k

1

x
1

1

e

e


x
a

k  1,2,  , n
Visual Surveillance System
Visual Surveillance System
Background Extraction

Non-recursive approaches
• Selective update using temporal averaging
• Selective update using temporal median
• Selective update using non-foreground pixels
• Non-parametric model
• Time Interval (It-L,It-L+1,It-1)
• Probability Density Function
1 t 1
f ( I t  u )   K (u  I i )
L i t  L
Background Extraction

Recursive approaches
• Kalman filter
• Mixture of Gaussians (MoG)
• Parametric model f ( I  u)   w
• Matching I    T  
• Updata w  (1   )w  
t
t
i, j
i , j 1
K
i 1
i, j
i , j 1
i , j 1
i , j  (1  p ) i , j 1  pI t
 i2, j  (1  p ) i2, j 1  p ( I t  i , j )
p    (u;  i , j ;  i , j )
 (u; i , j ;  i , j )
Improved Method

Treat the n×n block as a pixel
Improved Method(cont.)
Conclusions


Reduce Memory Size
Enhanced Performance
• Quantize the content of image
• Judgment of the variance