Physical Fluctuomatics
5th and 6th Probabilistic information processing
by Gaussian graphical model
Kazuyuki Tanaka
Graduate School of Information Sciences, Tohoku University
[email protected]
http://www.smapip.is.tohoku.ac.jp/~kazu/
Physical Fluctuomatics (Tohoku
University)
1
Textbooks
Kazuyuki Tanaka: Introduction of Image
Processing by Probabilistic Models,
Morikita Publishing Co., Ltd., 2006 (in
Japanese) , Chapter 7.
K. Tanaka: Statistical-mechanical
approach to image processing (Topical
Review), Journal of Physics A:
Mathematical and General, vol.35, no.37,
pp.R81-R150, 2002, Section 4.
Physical Fluctuomatics (Tohoku
University)
2
Contents
1.
2.
3.
4.
5.
Introduction
Probabilistic Image Processing
Gaussian Graphical Model
Statistical Performance Analysis
Concluding Remarks
Physical Fluctuomatics (Tohoku University)
3
Contents
1.
2.
3.
4.
5.
Introduction
Probabilistic Image Processing
Gaussian Graphical Model
Statistical Performance Analysis
Concluding Remarks
Physical Fluctuomatics (Tohoku University)
4
Markov Random Fields for Image Processing
Markov Random Fields
are One of Probabilistic Methods for Image processing.
S. Geman and D. Geman (1986): IEEE Transactions on PAMI
Image Processing for Markov Random Fields (MRF)
(Simulated Annealing, Line Fields)
J. Zhang (1992): IEEE Transactions on Signal Processing
Image Processing in EM algorithm for Markov
Random Fields (MRF) (Mean Field Methods)
Physical Fluctuomatics (Tohoku
University)
5
Markov Random Fields for Image Processing
In Markov Random Fields, we have to consider not
only the states with high probabilities but also ones
Hyperparameter
with low probabilities.
Estimation
In Markov Random Fields, we
have to estimate not only the
image but also hyperparameters
Statistical Quantities
in the probabilistic model.
We have to perform the
Estimation of Image
calculations of statistical
quantities repeatedly.
We can calculate statistical quantities by adopting the
Gaussian graphical model as a prior probabilistic model
and by using Gaussian integral formulas.
Physical Fluctuomatics (Tohoku
University)
6
Purpose of My Talk
Review of formulation of probabilistic
model for image processing by means of
conventional statistical schemes.
Review of probabilistic image processing
by using Gaussian graphical model
(Gaussian Markov Random Fields) as the
most basic example.
K. Tanaka: Statistical-Mechanical Approach
to Image Processing (Topical Review), J. Phys.
A: Math. Gen., vol.35, pp.R81-R150, 2002.
Section 2 and Section 4 are summarized in the present talk.
Physical Fluctuomatics (Tohoku
University)
7
Contents
1.
2.
3.
4.
5.
Introduction
Probabilistic Image Processing
Gaussian Graphical Model
Statistical Performance Analysis
Concluding Remarks
Physical Fluctuomatics (Tohoku University)
8
Bayes Formula and Bayesian Network
Pr{B | A} Pr{ A}
Pr{ A | B} 
Pr{B}
Prior
Probability
Data-Generating Process
Posterior Probability
Bayes Rule
Event B is given as the observed data.
Event A corresponds to the original
information to estimate.
Thus the Bayes formula can be applied to the
estimation of the original information from
the given data.
Physical Fluctuomatics (Tohoku
University)
A
B
Bayesian
Network
9
Image Restoration by Probabilistic Model
Noise
Transmission
Original
Image
Degraded
Image
Posterior



Pr{Original Image | Degraded Image }
Assumption 1: The degraded
image is randomly generated
from the original image by
according to the degradation
process.
Assumption 2: The original
image is randomly generated
by according to the prior
probability.
Prior
Likelihood


 


Pr{Degraded Image | Original Image }Pr{Original Image }

Pr{DegradedIm age}

Marginal Likelihood
Bayes Formula
Physical Fluctuomatics (Tohoku
University)
10
Image Restoration by Probabilistic Model
The original images and degraded images are represented
by f = (f1,f2,…,f|V|) and g = (g1,g2,…,g|V|), respectively.
Original
Degraded
Image
Image

ri  ( xi , yi )
Position Vector
i
of Pixel i
i
fi: Light Intensity of Pixel i
gi: Light Intensity of Pixel i
in Original Image
in Degraded Image
Physical Fluctuomatics (Tohoku
University)
11
Probabilistic Modeling of Image
Random Field of Original
Restoration
Image F  ( F1 , F2 ,..., F|V | )
Random Field of Degraded Image G  (G1 , G2 ,..., G|V | )
Posterior



Pr{Original Image | Degraded Image }
Prior
Likelihood


 


 Pr{Degraded Image | Original Image }Pr{Original Image }
Assumption 1: A given degraded image is obtained from the
original image by changing the state of each pixel to another
state by the same probability, independently of the other pixels.
State Vector of Original Image f  ( f1 , f 2 ,..., f |V | )
State Vector of Degraded Image g  ( g1 , g 2 ,..., g |V | )
Pr{Degraded Image g | Original Image f }
 Pr{G  g | F  f }    ( gi , fi )
Random Fields
iV
Physical Fluctuomatics (Tohoku
University)
gi
gi
or
fi
fi
12
Probabilistic Modeling of
Image Restoration Random Field of Original Image F  (F , F ,..., F
1
2
|V |
State Vector of Original Image f  ( f1 , f 2 ,..., f |V | )
Posterior



Pr{Original Image | Degraded Image }
Prior
Likelihood


 


 Pr{Degraded Image | Original Image }Pr{Original Image }
Assumption 2: The original image is generated according to a
prior probability. Prior Probability consists of a product of
functions defined on the neighbouring pixels.
Pr{Original Image f }
 Pr{F  f } 
 ( f i , f j )
ijE
i
j
Random Fields
Product over All the Nearest Neighbour Pairs of Pixels
Physical Fluctuomatics (Tohoku
University)
13
)
Bayesian Image Analysis
PrF  f 
f
Prior Probability
Original
Image
PrG  g F  f 
g
Degradation Process
Degraded
Image
g
Posterior Probability
PrF  f G  g 
V：Set of All
the pixels
E：Set of all
the nearest
neighbour
pairs of pixels
PrG  g F  f PrF  f 
PrG  g




    ( f i , g i )    ( f i , f j ) 
 iV
 {i , j}E

Image processing is reduced to calculations of
averages, variances and co-variances in the
posterior probability.
Physical Fluctuomatics (Tohoku
University)
14
Estimation of Original Image
We have some choices to estimate the restored image from
posterior probability.
In each choice, the computational time is generally
exponential order of the number of pixels.
(1) fˆ  arg max Pr{F  z | G  g}
z
(2) fˆi  arg max Pr{Fi  zi | G  g}
zi
Maximum A Posteriori (MAP)
estimation
Maximum posterior
marginal (MPM) estimation
Pr{Fi  f i | G  g} 
 Pr{F 
f | G  g}
f \ fi


ˆ  arg min   z Pr{F  z | G  g}dz 2
f
i  i
(3) i
i
Thresholded Posterior
Mean (TPM) estimation
Physical Fluctuomatics (Tohoku
University)
15
Contents
1.
2.
3.
4.
5.
Introduction
Probabilistic Image Processing
Gaussian Graphical Model
Statistical Performance Analysis
Concluding Remarks
Physical Fluctuomatics (Tohoku University)
16
Bayesian Image Analysis by
Gaussian Graphical Model
Prior
Probability
V:Set of all
the pixels
 1
1
2

PrF  f |   
exp      f i  f j  
Z PR ( )
 2 {i , j}E

 1
2



Z PR ( )      exp     zi  z j dz1dz2  dz|V |
 

 2 {i , j}E

 

Patterns are generated by MCMC.
  0.0001
E:Set of all the
nearest-neighbour
pairs of pixels
f i   ,
  0.0005
  0.0030
Markov Chain Monte Carlo Method
Physical Fluctuomatics (Tohoku
University)
17
Bayesian Image Analysis by
Gaussian Graphical Model
Degradation Process is assumed to be the
additive white Gaussian noise.
PrG  g F  f ,   
f i , gi   ,

1
2



exp

f

g

i
i 

2
2
 2

iV 2
1
V: Set of all
the pixels
Original Image f
Gaussian Noise n
Degraded Image g
Degraded image is
obtained by adding a
white Gaussian noise to
the original image.

g i  f i ~ N 0,  2

Histogram of Gaussian Random Numbers
Physical Fluctuomatics (Tohoku
University)
18
Bayesian Image Analysis
PrF  f 
f
Prior Probability
Original
Image
PrG  g F  f 
g
Degradation Process
Degraded
Image
g
Posterior Probability
PrF  f G  g 
V：Set of All
the pixels
E：Set of all
the nearest
neighbour
pairs of pixels
PrG  g F  f PrF  f 
PrG  g




    ( f i , g i )    ( f i , f j ) 
 iV
 {i , j}E

Image processing is reduced to calculations of
averages, variances and co-variances in the
posterior probability.
Physical Fluctuomatics (Tohoku
University)
19
Bayesian Image Analysis
A Posteriori Probability
f i , g j   ,
Pr{G  g | F  f ,  } Pr{F  f |  }
Pr{F  f | G  g ,  ,  } 
Pr{G  g |  ,  }







1

 1
2  
2 





  exp  
f

g
exp


f

f
 


i
i
i
j
2


 iV
 {i , j}E
2
2







 
 

 ( fi , gi )
 ( fi , f j )






    ( f i , g i )    ( f i , f j ) 
 iV
 {i , j}E

Physical Fluctuomatics (Tohoku
University)
20
Statistical Estimation of Hyperparameters
Hyperparameters ,  are determined so as to
maximize the marginal likelihood Pr{G=g|,} with
respect to , .
(ˆ , ˆ )  arg max Pr{G  g |  ,  }
 , 
fˆ   z Pr{F  z | G  g, ˆ , ˆ } dz
Pr{G  g |  ,  }   Pr{F  z , G  g |  ,  }dz
  Pr{G  g | F  z ,  } Pr{F  z |  }dz
y
V
x
Original Image
Pr{F  f |  }
f
Degraded Image
Pr{G  g | F  f ,  }
g
g
Marginalized with respect to F
Pr{G  g |  ,  }
Physical Fluctuomatics (Tohoku
University)
Marginal Likelihood
21
Bayesian Image Analysis
f i , g j   ,
A Posteriori Probability
Pr{F  f | G  g ,  ,  } 
Pr{G  g | F  f ,  } Pr{F  f |  }
Pr{G  g |  ,  }

1
1

exp  
2
Z POS  g ,  ,  
 2

 f
iV
 gi 
2
i
1
2
    f i  f j  
2 {i , j }E

1
1
1

T
T 
exp  
(
f

g
)(
f

g
)


fCf

2
Z POS  g ,  ,  
2
 2


1
T 1
T

Z POS  g,  ,        exp  
( z  g )( z  g )   z C z dz1dz 2  dz|V |
2
 

2
 2

 

Gaussian Graphical Model
f  ( f1 , f 2 ,..., f |V | )
g  ( g1 , g 2 ,..., g |V | )
z  ( z1 , z 2 ,..., z|V | )
|V|x|V| matrix
4

i C j   1
0

Physical Fluctuomatics (Tohoku
University)
i  j 
{i, j}  E 
otherwise 
22
Average of Posterior Probability
EF  ,    
 

 


  z Pr{F  z | G  g,  ,  }dz1 dz 2  dz|V |
T


1
I
I




2


z
exp
z

g
I


C
z

g

 dz1 dz 2  dz|V |
2
    2 2  I   2 C 
I   C  



T


 

1
I
I




2


exp
z

g
I


C
z

g

 dz1 dz 2  dz|V |
2
    2 2  I   2 C 
I   C  


I

g
I   2 C
 





|V |





 

 1
2
T
       ( z  m ) exp   2 ( z  m )( I   C )( z  m) dz1dz 2  dz|V |  (0,0,,0)
Gaussian Integral formula
Physical Fluctuomatics (Tohoku
University)
23
Bayesian Image Analysis
by Gaussian Graphical Model
f i , gi   ,
Posterior Probability
 1
Pr{F  f | G  g,  ,  }  exp   2
 2
 f
iV
 gi 
2
i
1
2
    f i  f j  
2 {i , j}E

Average of the posterior probability
can be calculated by using the multidimensional Gauss integral Formula
fˆ   z Pr{F  z | G  g,  ,  } dz

I
I   C
2
|V|x|V| matrix
g
V:Set of all
the pixels
E:Set of all the
nearest-neghbour
pairs of pixels
4

i C j   1
0

i  j 
{i, j}  E 
otherwise 
Multi-Dimensional Gaussian Integral Formula
Physical Fluctuomatics (Tohoku
University)
24
Statistical Estimation of Hyperparameters
Pr{G  g |  ,  }   Pr{G  g | F  z ,  } Pr{F  z |  }dz 
Z POS ( g,  ,  )
(2 2 )|V | / 2 Z PR ( )
Z POS  g ,  ,  

 

 
1

T 1
T
exp 
(
z

g
)(
z

g
)


z
C
z
dz1dz 2  dz|V |
2

2
 2



 1
T
Z PR        exp    z C z dz1dz 2  dz|V |
 

 2

 
y
V
x

Original Image
Pr{F  f |  }
f
Degraded Image
Pr{G  g | F  f ,  }
g
g
Marginalized with respect to F
Pr{G  g |  ,  }
Physical Fluctuomatics (Tohoku
University)
Marginal Likelihood
25
Calculations of Partition Function
(2 )|V |
 1

Z PR ( )      exp   z Cz dz1dz 2  dz|V | 
 

 2

 |V | det C
 

Z POS ( g ,  ,  )

1
     exp  
 2 2
 


 




I
 z 
g  I   2 C
I   2 C 

T



I
1
C
T
 z 
g   g
g dz1dz 2  dz|V |

2
2
2
I   C 
I   C



 1

C
 exp   g
g T 
I   2 C
 2


1
     exp  
 2 2
 


 





I
 z 
g  I   2 C
I   2 C 

T

 
I
 z 
g  dz1dz 2  dz|V |
2
I   C  


(2 2 )|V |
 1

C
exp   g
g T 
det( I   2 C )
I   2 C
 2

 1
T

exp

(
z

m
)
A
(
z

m
)

dz1dz 2  dz|V | 
       2

 

Gaussian Integral formula
(2 )|V |
det A
(A is a real symmetric and positive definite matrix.)
Physical Fluctuomatics (Tohoku
University)
26
Exact expression of Marginal Likelihood in
Gaussian Graphical Model
Pr{G  g |  ,  }   Pr{G  g | F  z ,  } Pr{F  z |  }dz 
Z POS ( g,  ,  )
(2 2 )|V | / 2 Z PR ( )
Multi-dimensional Gauss integral formula
det C 
 1
C
T

Pr{G  g |  ,  } 
exp   g
g 
I   2C
 2

2  V det I   2C


We can construct an exact EM algorithm.
ˆ ,ˆ   arg max Pr{G  g |  , }
 , 
fˆ 
I
I   C
2
g
Physical Fluctuomatics (Tohoku
University)
y
V
x
27
Bayesian Image Analysis
by Gaussian Graphical Model
Iteration Procedure in Gaussian Graphical Model
 1

 t  12 C
1
C
T

 t  
Tr

g
g
2
2
|V |

|
V
|
I   t  1 t  1 C
I   t  1 t  1 C



g
1

1
 t  12 I
1
 t  12  t  14 C 2
 t  
Tr

g
gT
|V |
I   t  1 t  12 C | V | I   t  1 t  12 C 2

m (t ) 

I
I   (t ) (t ) 2 C
g


f̂
fˆi (t )  arg min zi  mi (t ) 2
zi
t 
0.0008
0.001
0.0006
0.0004
0.0002
g
0
0
20
40
60
 100t 
80
Physical Fluctuomatics (Tohoku
University)
f̂
28
Image Restoration by Markov Random
Field Model and Conventional Filters
MSE
Original Image Degraded Image
Statistical Method
315
Lowpass
Filter
(3x3)
388
(5x5)
413
Median
Filter
(3x3)
486
(5x5)
445
MSE 


1
ˆ 2
f

f
 i i
| V | iV
V:Set of all
the pixels
Restored
Image
MRF
(3x3) Lowpass
(5x5) Median
Physical Fluctuomatics (Tohoku
University)
29
Contents
1.
2.
3.
4.
5.
Introduction
Probabilistic Image Processing
Gaussian Graphical Model
Statistical Performance Analysis
Concluding Remarks
Physical Fluctuomatics (Tohoku University)
30
Performance Analysis

x
Signal
5
 ˆ 2
1
[MSE ]   || x  x n ||
5 n 1

y1

y2

y3

y4

y5
Observed
Data
Physical Fluctuomatics (Tohoku
University)
Posterior Probability
Additive White
Gaussian Noise
Sample Average of Mean Square Error

x̂1

x̂ 2

x̂3

x̂ 4

x̂5
Estimated Results
31
Statistical Performance Analysis

1
MSE( |  , x ) 
V

 
   
2

h ( y,  ,  )  x Pr{Y  y | X  x,  } dy

x
Original Image

y
Degraded Image
   
Pr{Y  y | X  x ,  }
g
Additive White Gaussian Noise
 
h ( y,  ,  )
   
Pr{Y  y | X  x ,  }
   
Pr{ X  x | Y  y,  ,  }
Restored Image
Additive White Gaussian Noise
Physical Fluctuomatics (Tohoku
University)
Posterior Probability
32
Statistical Performance
Analysis

1
MSE( |  , x ) 
V
1

V

 I   2C 
   
 x P ( x | y ) dx




    
2
y  x Pr Y  y X  x dy
1 
 
h  y,  ,   

 
    
2
h  y,  ,    x Pr Y  y X  x dy

1 
2
 I   C
y
 4, i  j  V

i C j   1, {i, j}  E
 0, otherwise



   
Pr Y  y X  x ,  
1


exp


  2 2 xi  yi 2 
iV
 exp( 
Physical Fluctuomatics (Tohoku
University)
1
2 2
 2
xy )
33
Statistical Performance Estimation for
Gaussian Markov Random Fields

1
MSE ( |  , x ) 
V

1

V
I

1
V


I
 
   
2

h ( y,  ,  )  x  Pr{Y  y | X  x ,  } dy
  2 C
I
I   2 C
 
yx
2
 1
 
 2 
 
( y  x) 



|V |
1   2 

exp  
x  y dy
 2 2

2
   1
x  x 
I   2 C
 2 
I
2
 
 2 C   1
( y  x) 
x 
I   2 C
I   2 C
 2 
1

V
I

1

V

 
I
 2 C

(
y

x
)

  I   2C
I   2 C

x 

T
 1
x 
 2 
1   2 

exp  
y  x dy
 2 2

1   2 

exp  
y  x dy
 2 2




|V |
1   2 

exp  
y  x  dy
 2 2

|V |
1   2 

exp  
y  x  dy
 2 2

  1 
 2 C
( y  x )

( I   2 C ) 2
 2  
1

V

 xT
 2 4 C 2  1 
x

( I   2 C ) 2  2  
  1
x 
 2 



|V |
1   2 

exp  
y  x  dy
 2 2

1   2 

exp  
y  x  dy
 2 2

|V |

 xT

|V |
|V |



1

V
|V |







 
I
 2 C

(
y

x
)

 I   2 C
I   2 C

  1
1  
I
  ( y  x)T
( y  x )
V
( I   2 C ) 2
 2 
1  
 2 C
  ( y  x)T
V
( I   2 C ) 2
 4, i  j  V

i C j   1, {i, j}  E
 0, otherwise

=0
1   2 

exp  
y  x  dy
 2 2

 1   2 4 C 2  T
1 
 2I

Tr 
x
x
V  ( I   2 C ) 2  | V | ( I   2 C ) 2
Physical Fluctuomatics (Tohoku
University)
34
Statistical Performance Estimation for
Gaussian Markov Random Fields

1
MSE ( |  , x ) 
V

1

V
I

 
   
2

h ( y,  ,  )  x  Pr{Y  y | X  x ,  } dy
I   2 C
 
yx
2
 1 
 

2
 2  
|V |
1   2 

exp  
x  y dy
2
 2

 1  T  2 4 C 2 
1 
 2I

 Tr 
x
x
2
2
V  ( I   2 C ) 2  | V |
( I   C )

MSE ( |  , x )

MSE ( |  , x )
600
=40
600
400
400
200
200
0
0
0
0.001
0.002

0.003
 4, i  j  V

i C j   1, {i, j}  E
 0, otherwise

=40
0
Physical Fluctuomatics (Tohoku
University)
0.001
0.002

0.003
35
Contents
1.
2.
3.
4.
5.
Introduction
Probabilistic Image Processing
Gaussian Graphical Model
Statistical Performance Analysis
Concluding Remarks
Physical Fluctuomatics (Tohoku University)
36
Summary
Formulation of probabilistic model for
image processing by means of
conventional statistical schemes has been
summarized.
Probabilistic image processing by using
Gaussian graphical model has been shown
as the most basic example.
Physical Fluctuomatics (Tohoku
University)
37
References
K. Tanaka: Introduction of Image Processing by
Probabilistic Models, Morikita Publishing Co.,
Ltd., 2006 (in Japanese) .
K. Tanaka: Statistical-Mechanical Approach to
Image Processing (Topical Review), J. Phys. A,
35 (2002).
A. S. Willsky: Multiresolution Markov Models for
Signal and Image Processing, Proceedings of
IEEE, 90 (2002).
Physical Fluctuomatics (Tohoku
University)
38
Problem 5-1: Derive the expression of the
posterior probability Pr{F=f|G=g,,} by using
Bayes formulas Pr{F=f|G=g,,}
=Pr{G=g|F=f,}Pr{F=f,}/Pr{G=g|,}. Here
Pr{G=g|F=f,} and Pr{F=f,} are assumed to be
as follows:


1
1

PrG  g F  f ,    
exp  
f i  g i 2 
2
2
 2
2
 1
1
PrF  f |   
exp     f i  f j
Z PR ( )
 2 {i , j}E

iV

 
2


 1
2

dz1dz2  dz|V |



exp


z

z
i
j
  2 {i

 
, j }E

Z PR ( )  
 
[Answer]
Pr{F  f | G  g,  ,  } 

1
1
exp  
2
Z POS g,  ,  
 2

 1

exp
   2 2
 
Z PR    
 
 f
iV
i
1
2
2
 gi      f i  f j  
2 {i , j}E


1
2
2
 zi  gi   2   zi  z j  dz1dz2 dz|V |
iV
{i , j }E
Physical Fluctuomatics (Tohoku
University)

39
Problem 5-2: Show the following equality.
 f
1
2
2
iV
 gi 
2
i
1
1
2


   fi  f j
2 {i , j}E
1
T

(
f

g
)(
f

g
)


f
C
f
2 2
2
T
4

i C j   1
0

i  j 
{i, j}  E 
otherwise 
Physical Fluctuomatics (Tohoku
University)
40
Problem 5-3: Show the following equality.
1
1
T
( z  g )( z  g )   z C z
2
2
2
T


1 
I
2
 z 

g  I   C
2
2
2 
I   C 
1
C
T
 g
g
2
2
I   C

Physical Fluctuomatics (Tohoku
University)


I
 z 
g 
2
I   C 

T
41
Problem 5-4: Show the following equalities by using the
multi-dimensional Gaussian integral formulas.


1


exp

  2 2
   
Z POS g,  ,    

 

1
2
2


f

g

 i i 2    fi  f j  dz1dz2  dz|V |
iV
{i , j }E

(2 2 )|V |
C
 1
T
exp


g
g


det( I   2C )
I   2C
 2

 1
(2 )|V |
2
Z PR ( )      exp      f i  f j  dz1dz2  dz|V | 
 

 |V | det C
 2 {i , j}E

 

Physical Fluctuomatics (Tohoku
University)
42
Problem 5-5: Derive the extremum conditions for the
following marginal likelihood Pr{G=g|,} with respect to
the hyperparameters  and .
det C 
 1
C
T
Pr{G  g |  ,  } 
exp   g
g 
2
V
2
I   C
 2

2  det I   C


[Answer]
1
 2C
1
C

Tr

g
gT
 |V |
I   2 C | V | I   2 C
1


2

1
 2I
1
 2 4C 2

Tr

g
gT
|V |
I   2C | V | I   2C 2



Physical Fluctuomatics (Tohoku
University)

43
Problem 5-6: Derive the extremum conditions for the
following marginal likelihood Pr{G=g|,} with respect to
the hyperparameters  and .
det C 
 1
C
T
Pr{G  g |  ,  } 
exp   g
g 
2
V
2
I   C
 2

2  det I   C


[Answer]
1
 2C
1
C

Tr

g
gT
 |V |
I   2 C | V | I   2 C
1


2

1
 2I
1
 2 4C 2

Tr

g
gT
|V |
I   2C | V | I   2C 2



Physical Fluctuomatics (Tohoku
University)

44
Problem 5-7: Make a program that generate a degraded
image by the additive white Gaussian noise. Generate some
degraded images from a given standard images by setting
=10,20,30,40 numerically. Calculate the mean square error
(MSE) between the original image and the degraded image.
MSE 
1
 f i  g i 2

| V | iV
Original Image
Gaussian Noise
Degraded Image
K.Tanaka: Introduction of Image
Processing by Probabilistic Models,
Morikita Publishing Co., Ltd., 2006 .
Sample Program:
http://www.morikita.co.jp/soft/84661/
Histogram of Gaussian Random
Numbers Fi Gi~N(0,402)
Physical Fluctuomatics (Tohoku
University)
45
Problem 5-8: Make a program of the following procedure in
probabilistic image processing by using the Gaussian
graphical model and the additive white Gaussian noise.
Algorithm:
Repeat the following procedure until convergence
 1

 t  12 C
1
C
T

 t  
Tr

g
g
2
2
|V |

I   t  1 t  1 C | V | I   t  1 t  1 C



1

1
 t  12 I
1
 t  12  t  14 C 2
 t  
Tr

g
gT
|V |
I   t  1 t  12 C | V | I   t  1 t  12 C 2

m (t ) 

I
I   (t ) (t ) 2 C
g


fˆi (t )  arg min zi  mi (t ) 2
zi
K.Tanaka: Introduction of Image Processing by Probabilistic Models,
Morikita Publishing Co., Ltd., 2006 .
Sample Program: http://www.morikita.co.jp/soft/84661/
Physical Fluctuomatics (Tohoku
University)
46