3 September, 2009 SSP2009, Cardiff, UK 1 Probabilistic Image Processing by Extended Gauss-Markov...

3 September, 2009 SSP2009, Cardiff, UK 1

Probabilistic Image Processingby Extended Gauss-Markov Random Fields

Kazuyuki TanakaKazuyuki Tanaka, Muneki YasudaMuneki Yasuda, Nicolas MorinNicolas Morin

Graduate School of Information Sciences, Tohoku University, Japanand

D. M. TitteringtonDepartment of Statistics, University of Glasgow, UKDepartment of Statistics, University of Glasgow, UK


Image Restoration by Bayesian Statistics

PriorProcessn Degradatio

Posterior

}Image OriginalPr{}Image Original|Image DegradedPr{

}Image Degraded|Image OriginalPr{

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


Bayesian Image Analysis

0005.0 0030.00001.0

Prior Probability

T

},{

2

},{

2 ))((2

1exp)(

2

1exp)(

2

1exp}|Pr{

21

xxxxxxxXEji

jiEji

ji

CI

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

Prior

Likelihood

Posterior

}Image OriginalPr{

}Image Original|Image DegradedPr{


otherwise,0

},{,

},{,1

,44

)(2

1

Eji

Eji

Vji

ji

C0

0

45.0

Gibbs Sampler



V:Set of all the pixels

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

Prior

Likelihood

Posterior

}Image OriginalPr{

}Image Original|Image DegradedPr{


22

22 2

1exp)(

2

1exp},|Pr{ yxyxxXyY

Viii

0



x

g

},|Pr{ xX

},|Pr{ xXyY

yOriginal

Image

Degraded Image

Prior Probability

Posterior Probability

Degradation Process

T22

},{

2

},{

222

)(2

1

2

1exp

)(2

1)(

2

1)(

2

1exp

},,|Pr{

},,Pr{},|Pr{

},,,|Pr{

21

xxyx

xxxxyx

yY

xXxXyY

yYxX

Ejiji

Ejiji

Viii

C

xdyYxXxx ii

},,,|Pr{ˆ Model Field Random Markov Gauss

),(

iX

SmoothingData Dominant

Bayesian Network Estimate

otherwise,0

},{,

},{,1

,44

)(2

1

Eji

Eji

Vji

ji

C


Average of Posterior Probability

)0,00()))(()((2

1exp)(

||

||21T2

V

V ,,dzdzdzzzz

CI

T22

||21T

2T2

22

||21T

2T2

22

||21

)()(

)()(2

1exp

)()(

2

1exp

},,|Pr{,,ˆ

yy

dzdzdzyzyz

dzdzdzyzyzz

dzdzdzyYzXzXx

V

V

V

CI

I

CI

I

CI

ICI

CI

I

CI

ICI

CI

I

Gaussian Integral formula

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


Degraded Image

Statistical Estimation of Hyperparameters

zdzXxXyYyY

},|Pr{},|Pr{},,|Pr{

},,|Pr{max arg)ˆ,ˆ(

,

yY

x

g

Marginalized with respect to X

},|Pr{ xX

},|Pr{ xXyY

yOriginal Image

Marginal Likelihood

},,|Pr{ yY

Hyperparameters are determinedso as to maximize the marginal likelihood Pr{Y=y|,,} with respect to ,


Statistical Estimation of Hyperparameters

)()2(

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

PR2/||2

POS

Z

yZzdzXzXyYyY

V

AA

det

)2()( )(

2

1exp

||

||21T

V

Vdzdzdzzz

T22

||2

||21T

2T2

22

T2

||21T

2T

2T2

22

POS

)(

)(

2

1exp

))(det(

)2(

)()(

)(

2

1exp

)(

)(

2

1exp

)(

)(

2

1

)()(

)(

2

1exp

),,,(

yy

dzdzdzyzyz

yy

dzdzdzyyyzyz

yZ

V

V

V

CI

C

CI

CI

ICI

CI

I

CI

C

CI

C

CI

ICI

CI

I

Gaussian Integral formula

)(det

)2()(

2

1exp)(

||

||

||21T

CC

V

V

VPR dzdzdzzzZ


Exact Expression of Marginal Likelihood in Gaussian Graphical Model

T

22 )(

)(

2

1exp

)(det2

)(det},,|Pr{ yyyY

V

CI

C

CI

C

1

T22

2

)(

)(

||

1

)(

)(Tr

||

1

yy

VV

CI

C

CI

C

T22

242

2

2

)(

)(

||

1

)(Tr

||

1yy

VV

CI

C

CI

I

Extremum Conditions for and

1

T22

2

)(11

)(

||

1

)(11

)(1Tr

||

1

y

tty

Vtt

t

Vt

CI

C

CI

C

T22

242

2

2

)(11

)(11

||

1

)(11

1Tr

||

1y

tt

tty

Vtt

t

Vt

CI

C

CI

I

),,|Pr{max arg)ˆ,ˆ(

,

yY

y

,,,

1,1

ttQ

tt

Iterated Algorithm

EM Algorithm

0},,|Pr{ ,0},,|Pr{

yYyY


Bayesian Image Analysis by Gaussian Graphical Model

y

Iteration Procedure of EM Algorithm in Gaussian Graphical Model

EM

x̂

y

),,|Pr{max arg)ˆ,ˆ(

,

yY

y

,,,1,1 ttQtt

x̂

T12 ))()()(()( yCI tttx


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

2||ˆ||

||

1MSE

Vi

xxV

MSEMSE

Conventional GMRF Conventional GMRF =0=0 309309

ExtendedExtendedGMRFGMRF

273273

231231

Simultaneous ARSimultaneous AR 225225

Extended GMRFExtended GMRF

Conventional Conventional GMRF GMRF =0=0

Original ImageOriginal Image Degraded ImageDegraded Image

RestoredRestoredImageImage

Extended GMRFExtended GMRF Simultaneous ARSimultaneous AR

(0.00088,33) (0.00465,36) (0.00171,38)2)(0.00051,3)ˆˆ( σ,α


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

2||ˆ||

||

1MSE

Vi

xxV

MSEMSE

Conventional GMRF Conventional GMRF =0=0 313313

ExtendedExtendedGMRFGMRF

310310

324324

Simultaneous ARSimultaneous AR 383383


Conventional Conventional GMRF GMRF =0=0

Original ImageOriginal Image Degraded ImageDegraded Image

RestoredRestoredImageImage


Simultaneous ARSimultaneous AR

(0.00123,39) (0.00687,41) (0.00400,43)8)(0.00073,3)ˆˆ( σ,α


Statistical Performance by Sample Average of Numerical Experiments

1y

x

2y

3y

4y

5y

1h

2h

3h

4h

5h

Pos

teri

or P

rob

abil

ity

Restored ImagesDegraded Images

Sample Average of Mean Square Error

Original Images

Noi

se

5

1

2||||5

1),(

nnhxE


Statistical Performance Estimation

ydxXyYxyhV

xE

},|Pr{),,,(1

)|,,(2

),,,( yh

g

y

Additive White Gaussian Noise

},|Pr{ xXyY

x

},,,|Pr{ yYxX

Posterior Probability

Restored Image

Original Image Degraded Image

},|Pr{ xXyY

Additive White Gaussian Noise


Statistical Performance Estimation for Gauss Markov Random Fields

T22

242

22

2

22

||T

22

242

22

||T

22

2

22

||T

22

2

22

||T

22

22

||T

2

2T

22

2

2

22

||2

2

2

2

22

||2

22

22

||2

2

2

))((

)(

||

1

))((Tr

1

2

1exp

2

1

))((

)(1

2

1exp

2

1)(

))((

)(1

2

1exp

2

1

))((

)()(

1

2

1exp

2

1)(

))(()(

1

2

1exp

2

1

)(

)()(

)()(

)(

)()(

1

2

1exp

2

1

)(

)(

)()(

1

2

1exp

2

1

)()()(

1

2

1exp

2

1

)(

1

},|Pr{),,,(1

)|,,(

xI

xVV

ydxyxxV

ydxyxyxV

ydxyxxyV

ydxyxyxyV

ydxyxxyxxyV

ydxyxxyV

ydxyxxxyV

ydyxxyV

ydxXyYxyhV

xE

V

V

V

V

V

V

V

V

C

C

CI

I

CI

C

CI

C

CI

C

CI

I

CI

C

CI

I

CI

C

CI

I

CI

C

CI

I

CI

I

CI

I

CI

I

= 0


200

250

300

350

400

0 0.005 0.01 0.015

200

250

300

350

400

0 0.005 0.01 0.015

Statistical Performance Estimation for Gauss Markov Random Fields

T22

242

22

2

22

||

2TT

22

2

))((

)(

||

1

))((Tr

1

2

1exp

2

1

)()(

1

},|Pr{),,,(1

)|,,(

xI

xVV

ydyxxyxyV

ydxXyYxyhV

xE

V

C

C

CI

I

CI

I

CI

I

=40=40

)|,,( xE )|,,( xE

otherwise,0

},{,

},{,1

,44

)(2

1

Eji

Eji

Vji

ji

C


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.


ReferenceReferencess

1.1. K. TanakaK. Tanaka: Statistical-mechanical approach to image processing (Topical Revie: Statistical-mechanical approach to image processing (Topical Review), Journal of Physics A: Mathematical and General, vol.35, no.37, pp.R81-R15w), Journal of Physics A: Mathematical and General, vol.35, no.37, pp.R81-R150, 2002.0, 2002.

2.2. K. TanakaK. Tanaka and J. Inoue: Maximum Likelihood Hyperparameter Estimation for and J. Inoue: Maximum Likelihood Hyperparameter Estimation for Solvable Markov Random Field Model in Image Restoration, IEICE TransactioSolvable Markov Random Field Model in Image Restoration, IEICE Transactions on Information and Systems, vol.E85-D, no.3, pp.546-557, 2002.ns on Information and Systems, vol.E85-D, no.3, pp.546-557, 2002.

3.3. K. TanakaK. Tanaka, J. Inoue and , J. Inoue and D. M. TitteringtonD. M. Titterington: Probabilistic Image Processing by : Probabilistic Image Processing by Means of Bethe Approximation for Q-Ising Model, Journal of Physics A: MatheMeans of Bethe Approximation for Q-Ising Model, Journal of Physics A: Mathematical and General, vol. 36, no. 43, pp.11023-11036, 2003.matical and General, vol. 36, no. 43, pp.11023-11036, 2003.

4.4. K. TanakaK. Tanaka, H. Shouno, M. Okada and , H. Shouno, M. Okada and D. M. TitteringtonD. M. Titterington: Accuracy of the Bethe : Accuracy of the Bethe Approximation for Hyperparameter Estimation in Probabilistic Image ProcessinApproximation for Hyperparameter Estimation in Probabilistic Image Processing, Journal of Physics A: Mathematical and General, vol.37, no.36, pp.8675-8696, g, Journal of Physics A: Mathematical and General, vol.37, no.36, pp.8675-8696, 2004.2004.

5.5. K. TanakaK. Tanaka and and D. M. TitteringtonD. M. Titterington: Statistical Trajectory of Approximate EM Al: Statistical Trajectory of Approximate EM Algorithm for Probabilistic Image Processing, Journal of Physics A: Mathematical gorithm for Probabilistic Image Processing, Journal of Physics A: Mathematical and Theoretical, vol.40, no.37, pp.11285-11300, 2007.and Theoretical, vol.40, no.37, pp.11285-11300, 2007.

3 September, 2009 SSP2009, Cardiff, UK 1 Probabilistic Image Processing by Extended Gauss-Markov...

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