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Probabilistic Image Processing by Extended Gauss-Markov Random Fields Kazuyuki Tanaka, Kazuyuki Tanaka Muneki Yasuda Yasuda, Nicolas Morin Nicolas Morin Muneki Graduate School of Information Sciences, Tohoku University, Japan and D. M.

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3 September, 2009 SSP2009, Cardiff, UK 1

Probabilistic Image Processing by Extended Gauss-Markov Random Fields

Kazuyuki Tanaka Kazuyuki Tanaka, Muneki Muneki Yasuda Yasuda, Nicolas Morin Nicolas Morin Graduate School of Information Sciences, Tohoku University, Japan and

  • D. M. Titterington

Department of Statistics, University of Glasgow, UK Department of Statistics, University of Glasgow, UK

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3 September, 2009 SSP2009, Cardiff, UK 2

Image Restoration by Bayesian Statistics

4 4 4 8 4 4 4 7 6 4 4 4 4 4 4 4 8 4 4 4 4 4 4 4 7 6 4 4 4 4 4 4 4 8 4 4 4 4 4 4 4 7 6

Prior Process n Degradatio Posterior

} Image Original Pr{ } Image Original | Image Degraded Pr{ } Image Degraded | Image Original Pr{ ∝

Assumption 1: Original images are randomly generated by according to a prior probability. Bayes Formula Assumption 2: Degraded images are randomly generated from the original image by according to the conditional probability of degradation process.

Original Image Degraded Image

Transmission

Noise Estimate

Posterior

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3 September, 2009 SSP2009, Cardiff, UK 3

Bayesian Image Analysis

0005 . = α 0030 . = α 0001 . = α

Prior Probability

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ∝ =

∏ ∏

∈ ∈ T } , { 2 } , { 2

)) ( ( 2 1 exp ) ( 2 1 exp ) ( 2 1 exp } | Pr{

2 1

x x x x x x x X

E j i j i E j i j i

r r r r γ α α αγ α α C I

Assumption 1: Prior Probability consists of a product of functions defined on the neighbouring pixels.

4 4 4 8 4 4 4 7 6 4 4 4 4 4 4 4 8 4 4 4 4 4 4 4 7 6 4 4 4 4 4 4 4 8 4 4 4 4 4 4 4 7 6

Prior Likelihood Posterior

} Image Original Pr{ } Image Original | Image Degraded Pr{ } Image Degraded | Image Original Pr{ × ∝

⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ∈ − ∈ − ∈ = + =

  • therwise

, } , { , } , { , 1 , 4 4 ) (

2 1

E j i E j i V j i j i γ γ γ C

> α

= γ 45 . − = γ

Gibbs Sampler

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3 September, 2009 SSP2009, Cardiff, UK 4

Bayesian Image Analysis

V:Set of all the pixels

Assumption 2: Degraded image is generated from the original image by Additive White Gaussian Noise.

4 4 4 8 4 4 4 7 6 4 4 4 4 4 4 4 8 4 4 4 4 4 4 4 7 6 4 4 4 4 4 4 4 8 4 4 4 4 4 4 4 7 6

Prior Likelihood Posterior

} Image Original Pr{ } Image Original | Image Degraded Pr{ } Image Degraded | Image Original Pr{ × ∝

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ∝ = =

∏

∈ 2 2 2 2

2 1 exp ) ( 2 1 exp } , | Pr{ y x y x x X y Y

V i i i

r r r r r r σ σ σ

> σ

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3 September, 2009 SSP2009, Cardiff, UK 5

Bayesian Image Analysis

x r

g

} , | Pr{ γ α x X r r =

} , | Pr{ σ x X y Y r r r r = =

y r

Original Image

Degraded Image

Prior Probability Posterior Probability Degradation Process

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − − − − − − ∝ = = = = = = =

∑ ∑ ∑

∈ ∈ ∈ T 2 2 } , { 2 } , { 2 2 2

) ( 2 1 2 1 exp ) ( 2 1 ) ( 2 1 ) ( 2 1 exp } , , | Pr{ } , , Pr{ } , | Pr{ } , , , | Pr{

2 1

x x y x x x x x y x y Y x X x X y Y y Y x X

E j i j i E j i j i V i i i

r r r r r r r r r r r r r r r r γ α σ αγ α σ σ γ α γ α σ σ γ α C

∫

= = = x d y Y x X x x

i i

r r r r r } , , , | Pr{ ˆ σ γ α

Model Field Random Markov Gauss ) , ( ⇒ +∞ −∞ ∈

i

X

Smoothing Data Dominant Bayesian Network

Estimate

⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ∈ − ∈ − ∈ = + =

  • therwise

, } , { , } , { , 1 , 4 4 ) (

2 1

E j i E j i V j i j i γ γ γ C

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3 September, 2009 SSP2009, Cardiff, UK 6

Average of Posterior Probability

) , ( ) ))( ( )( ( 2 1 exp ) (

| | | | 2 1 T 2

4 8 4 7 6 L L r r r r r r L

V V

, , dz dz dz z z z = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − − −

∫ ∫ ∫

∞ + ∞ − ∞ + ∞ − ∞ + ∞ −

μ γ ασ μ μ C I

( ) ( )

T 2 2 | | 2 1 T 2 T 2 2 2 | | 2 1 T 2 T 2 2 2 | | 2 1

) ( ) ( ) ( ) ( 2 1 exp ) ( ) ( 2 1 exp } , , | Pr{ , , ˆ y y dz dz dz y z y z dz dz dz y z y z z dz dz dz y Y z X z X x

V V V

r r L r r r r L L r r r r r L L r r r r r L r r γ ασ γ ασ γ ασ ασ γ ασ σ γ ασ ασ γ ασ σ σ α σ γ α C I I C I I C I I C I C I I C I I C I C I I + = + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − − = = = = =

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

∞ + ∞ − ∞ + ∞ − ∞ + ∞ − ∞ + ∞ − ∞ + ∞ − ∞ + ∞ − +∞ ∞ − +∞ ∞ − +∞ ∞ −

Gaussian Integral formula Average of the posterior probability can be calculated by using the multi-dimensional Gauss integral Formula

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3 September, 2009 SSP2009, Cardiff, UK 7

Degraded Image

Statistical Estimation of Hyperparameters

∫

= = = = = z d z X x X y Y y Y r r r r r r r r r } , | Pr{ } , | Pr{ } , , | Pr{ γ α σ σ γ α

( )

} , , | Pr{ max arg ) ˆ , ˆ (

,

σ γ α σ α

σ α

y Y r r = =

x r

g

Marginalized with respect to X

} , | Pr{ γ α x X r r =

} , | Pr{ σ x X y Y r r r r = =

y r

Original Image

Marginal Likelihood

} , , | Pr{ σ γ α y Y r r =

Hyperparameters α, σ are determined so as to maximize the marginal likelihood Pr{Y=y|α,γ,σ} with respect to α, σ.

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3 September, 2009 SSP2009, Cardiff, UK 8

Statistical Estimation of Hyperparameters

) ( ) 2 ( ) , , , ( } , | Pr{ } , | Pr{ } , , | Pr{

PR 2 / | | 2 POS

σ πσ σ γ α γ α σ σ γ α Z y Z z d z X z X y Y y Y

V

r r r r r r r r r r = = = = = =

∫

A A det ) 2 ( ) ( ) ( 2 1 exp

| | | | 2 1 T V V

dz dz dz z z π μ μ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − −

∫ ∫ ∫

∞ + ∞ − ∞ + ∞ − ∞ + ∞ −

L r r r r L

( ) ( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − − × ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − − =

∫ ∫ ∫ ∫ ∫ ∫

∞ + ∞ − ∞ + ∞ − ∞ + ∞ − ∞ + ∞ − ∞ + ∞ − ∞ + ∞ − T 2 2 | | 2 | | 2 1 T 2 T 2 2 2 T 2 | | 2 1 T 2 T 2 T 2 2 2 POS

) ( ) ( 2 1 exp )) ( det( ) 2 ( ) ( ) ( ) ( 2 1 exp ) ( ) ( 2 1 exp ) ( ) ( 2 1 ) ( ) ( ) ( 2 1 exp ) , , , ( y y dz dz dz y z y z y y dz dz dz y y y z y z y Z

V V V

r r L r r r r L r r L r r r r r r L r γ ασ γ α γ ασ πσ γ ασ γ ασ γ ασ σ γ ασ γ α γ ασ γ α γ ασ γ ασ γ ασ σ σ γ α C I C C I C I I C I C I I C I C C I C C I I C I C I I

Gaussian Integral formula

) ( det ) 2 ( ) ( 2 1 exp ) (

| | | | | | 2 1 T

γ α π γ α α C C

V V V PR

dz dz dz z z Z = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛− ≡ ∫

∫ ∫

∞ + ∞ − ∞ + ∞ − ∞ + ∞ −

L r r L

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3 September, 2009 SSP2009, Cardiff, UK 9

Exact Expression of Marginal Likelihood in Gaussian Graphical Model

( ) ( )

( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + = =

T 2 2

) ( ) ( 2 1 exp ) ( det 2 ) ( det } , , | Pr{ y y y Y

V

r r r r γ ασ γ α γ ασ π γ α σ γ α C I C C I C

( )

1 T 2 2 2

) ( ) ( | | 1 ) ( ) ( Tr | | 1

−

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + = y y V V r r γ ασ γ γ ασ γ σ α C I C C I C

( ) ( )

T 2 2 2 4 2 2 2

) ( ) ( | | 1 ) ( Tr | | 1 y y V V r r γ ασ γ σ α γ ασ σ σ C I C C I I + + + =

Extremum Conditions for α and σ

( ) ( ) ( ) ( )

( )

( ) ( )

1 T 2 2 2

) ( 1 1 ) ( | | 1 ) ( 1 1 ) ( 1 Tr | | 1

−

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − + + − − + − ← y t t y V t t t V t r r γ σ α γ γ σ α γ σ α C I C C I C

( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( )

( )

T 2 2 2 4 2 2 2

) ( 1 1 ) ( 1 1 | | 1 ) ( 1 1 1 Tr | | 1 y t t t t y V t t t V t r r γ σ α γ σ α γ σ α σ σ C I C C I I − − + − − + − − + − ←

( )

) , , | Pr{ max arg ) ˆ , ˆ (

,

σ γ α σ α

σ α

y Y r r = =

( ) ( ) ( ) ( ) ( ) ( )

y r , , , 1 , 1 γ σ α σ α t t Q t t ← + +

Iterated Algorithm EM Algorithm

} , , | Pr{ , } , , | Pr{ = = ∂ ∂ = = ∂ ∂ σ γ α σ σ γ α α y Y y Y

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3 September, 2009 SSP2009, Cardiff, UK 10

Bayesian Image Analysis by Gaussian Graphical Model

y r

Iteration Procedure of EM Algorithm in Gaussian Graphical Model EM

x ˆ r

y r

( )

) , , | Pr{ max arg ) ˆ , ˆ (

,

σ γ α σ α

σ α

y Y r r = = ( ) ( ) ( ) ( ) ( ) ( )

y r , , , 1 , 1 γ σ α σ α t t Q t t ← + +

x ˆ r

T 1 2

)) ( ) ( ) ( ( ) ( y C I r r

−

+ = γ σ α t t t x

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3 September, 2009 SSP2009, Cardiff, UK 11

Image Restoration by Gaussian Markov Random Field (GMRF) Model and Conventional Filters

2

|| ˆ || | | 1 MSE

∑

∈

− =

V i

x x V r r

309 309 Conventional GMRF Conventional GMRF γ γ=0 =0 225 225 Simultaneous AR Simultaneous AR 231 231 γ= γ=− −0.45 0.45 273 273 γ= γ=− −0.2 0.2 Extended Extended GMRF GMRF MSE MSE

Extended GMRF Extended GMRF γ= γ=− −0.2 0.2

Conventional Conventional GMRF GMRF γ γ=0 =0 Original Image Original Image Degraded Image Degraded Image Restored Restored Image Image

Extended GMRF Extended GMRF γ= γ=− −0.45 0.45 Simultaneous AR Simultaneous AR

(0.00088,33) (0.00465,36) (0.00171,38)

2) (0.00051,3 ) ˆ ˆ ( = σ , α

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3 September, 2009 SSP2009, Cardiff, UK 12

Image Restoration by Gaussian Markov Random Field (GMRF) Model and Conventional Filters

2

|| ˆ || | | 1 MSE

∑

∈

− =

V i

x x V r r

313 313 Conventional GMRF Conventional GMRF γ γ=0 =0 383 383 Simultaneous AR Simultaneous AR 324 324 γ= γ=− −0.45 0.45 310 310 γ= γ=− −0.2 0.2 Extended Extended GMRF GMRF MSE MSE

Extended GMRF Extended GMRF γ= γ=− −0.2 0.2

Conventional Conventional GMRF GMRF γ γ=0 =0 Original Image Original Image Degraded Image Degraded Image Restored Restored Image Image

Extended GMRF Extended GMRF γ= γ=− −0.45 0.45 Simultaneous AR Simultaneous AR

(0.00123,39) (0.00687,41) (0.00400,43)

8) (0.00073,3 ) ˆ ˆ ( = σ , α

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3 September, 2009 SSP2009, Cardiff, UK 13

Statistical Performance by Sample Average of Numerical Experiments

1

y r

x r

2

y r

3

y r

4

y r

5

y r

1

h r

2

h r

3

h r

4

h r

5

h r

Posterior Probability

Restored Images Degraded Images

Sample Average of Mean Square Error

Original Images

Noise

∑

=

− ≅

5 1 2

|| || 5 1 ) , (

n n

h x E r r σ α

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3 September, 2009 SSP2009, Cardiff, UK 14

Statistical Performance Estimation

y d x X y Y x y h V x E r r r r r r r r r } , | Pr{ ) , , , ( 1 ) | , , (

2

σ σ γ α σ γ α = = − =

∫

) , , , ( σ γ α y h r r

g

y r

Additive White Gaussian Noise

} , | Pr{ σ x X y Y r r r r = =

x r

} , , , | Pr{ σ γ α y Y x X r r r r = =

Posterior Probability

Restored Image Original Image Degraded Image

} , | Pr{ σ x X y Y r r r r = =

Additive White Gaussian Noise

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3 September, 2009 SSP2009, Cardiff, UK 15

Statistical Performance Estimation for Gauss Markov Random Fields

T 2 2 2 4 2 2 2 2 2 2 | | T 2 2 2 4 2 2 2 | | T 2 2 2 2 2 | | T 2 2 2 2 2 | | T 2 2 2 2 | | T 2 2 T 2 2 2 2 2 2 | | 2 2 2 2 2 2 | | 2 2 2 2 2 | | 2 2 2

)) ( ( ) ( | | 1 )) ( ( Tr 1 2 1 exp 2 1 )) ( ( ) ( 1 2 1 exp 2 1 ) ( )) ( ( ) ( 1 2 1 exp 2 1 )) ( ( ) ( ) ( 1 2 1 exp 2 1 ) ( )) ( ( ) ( 1 2 1 exp 2 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 2 1 exp 2 1 ) ( ) ( ) ( ) ( 1 2 1 exp 2 1 ) ( ) ( ) ( 1 2 1 exp 2 1 ) ( 1 } , | Pr{ ) , , , ( 1 ) | , , ( x I x V V y d x y x x V y d x y x y x V y d x y x x y V y d x y x y x y V y d x y x x y x x y V y d x y x x y V y d x y x x x y V y d y x x y V y d x X y Y x y h V x E

V V V V V V V V

r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r γ ασ γ σ α γ ασ σ σ σ π γ ασ γ σ α σ σ π γ ασ γ ασ σ σ π γ ασ γ ασ σ σ π γ ασ σ σ π γ ασ γ ασ γ ασ γ ασ γ ασ γ ασ σ σ π γ ασ γ ασ γ ασ σ σ π γ ασ γ ασ σ σ π γ ασ σ σ γ α σ γ α C C C I I C I C C I C C I C C I I C I C C I I C I C C I I C I C C I I C I I C I I C I I + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + + + − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ × − + = = = × − =

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

= 0

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3 September, 2009 SSP2009, Cardiff, UK 16

200 250 300 350 400 0.005 0.01 0.015

200 250 300 350 400 0.005 0.01 0.015

Statistical Performance Estimation for Gauss Markov Random Fields

T 2 2 2 4 2 2 2 2 2 2 | | 2 T T 2 2 2

)) ( ( ) ( | | 1 )) ( ( Tr 1 2 1 exp 2 1 ) ( ) ( 1 } , | Pr{ ) , , , ( 1 ) | , , ( x I x V V y d y x x y x y V y d x X y Y x y h V x E

V

r r r r r r r r r r r r r r r r r r γ ασ γ σ α γ ασ σ σ σ π γ ασ γ ασ σ σ γ α σ γ α C C C I I C I I C I I + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ × ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + = = = × − =

∫ ∫

σ=40 σ=40 α

) | , , ( x E r σ γ α ) | , , ( x E r σ γ α

⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ∈ − ∈ − ∈ = + =

  • therwise

, } , { , } , { , 1 , 4 4 ) (

2 1

E j i E j i V j i j i γ γ γ C

γ=0

γ=0 γ=−0.2 γ=−0.45

α

γ=−0.2

γ=−0.45

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3 September, 2009 SSP2009, Cardiff, UK 17

Summary

We propose an extension of the Gauss-Markov random field models by introducing next-nearest neighbour

  • interactions. Values for the hyperparameters in the

proposed model are determined by using the EM algorithm in order to maximize the marginal likelihood. In addition, a measure of mean squared error, which quantifies the statistical performance of our proposed model, is derived analytically as an exact explicit expression by means of the multi-dimensional Gaussian integral formulas. Statistical performance analysis of probabilistic image processing for our extended Gauss Markov Random Fields has been shown.

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3 September, 2009 SSP2009, Cardiff, UK 18

References References

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: Statistical-

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R150, 2002. 2. 2.

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D, no.3, pp.546-

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, J. Inoue and D. M.

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