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
3 September, 2009 SSP2009, Cardiff, UK 2
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
3 September, 2009 SSP2009, Cardiff, UK 3
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{
}Image Degraded|Image OriginalPr{
otherwise,0
},{,
},{,1
,44
)(2
1
Eji
Eji
Vji
ji
C0
0
45.0
Gibbs Sampler
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.
Prior
Likelihood
Posterior
}Image OriginalPr{
}Image Original|Image DegradedPr{
}Image Degraded|Image OriginalPr{
22
22 2
1exp)(
2
1exp},|Pr{ yxyxxXyY
Viii
0
3 September, 2009 SSP2009, Cardiff, UK 5
Bayesian Image Analysis
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
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xdyYxXxx ii
},,,|Pr{ˆ Model Field Random Markov Gauss
),(
iX
SmoothingData Dominant
Bayesian Network Estimate
otherwise,0
},{,
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,44
)(2
1
Eji
Eji
Vji
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C
3 September, 2009 SSP2009, Cardiff, UK 6
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
3 September, 2009 SSP2009, Cardiff, UK 7
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 ,
3 September, 2009 SSP2009, Cardiff, UK 8
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
3 September, 2009 SSP2009, Cardiff, UK 9
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
3 September, 2009 SSP2009, Cardiff, UK 10
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
3 September, 2009 SSP2009, Cardiff, UK 11
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
3 September, 2009 SSP2009, Cardiff, UK 12
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
Extended GMRFExtended GMRF
Conventional Conventional GMRF GMRF =0=0
Original ImageOriginal Image Degraded ImageDegraded Image
RestoredRestoredImageImage
Extended GMRFExtended GMRF
Simultaneous ARSimultaneous AR
(0.00123,39) (0.00687,41) (0.00400,43)8)(0.00073,3)ˆˆ( σ,α
3 September, 2009 SSP2009, Cardiff, UK 13
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
3 September, 2009 SSP2009, Cardiff, UK 14
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
3 September, 2009 SSP2009, Cardiff, UK 15
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
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22
||T
2
2T
22
2
2
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||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
3 September, 2009 SSP2009, Cardiff, UK 16
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
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.
3 September, 2009 SSP2009, Cardiff, UK 18
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.
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