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1

Bayesian Image Modeling by Generalized Sparse Markov Random Fields and Loopy Belief Propagation

Kazuyuki TanakaGSIS, Tohoku University, Sendai, Japanhttp://www.smapip.is.tohoku.ac.jp/~kazu/

Collaborators

Muneki Yasuda (GSIS, Tohoku University, Japan

)Sun Kataoka (GSIS, Tohoku University, Japan)D. M. Titterington (Department of Statistics, University of Glasgow, UK)

20 March, 2013

SPDSA2013, Sendai, Japan

Slide22

Outline

Supervised Learning of Pairwise Markov Random Fields by Loopy Belief PropagationBayesian Image Modeling by Generalized Sparse PriorNoise Reductions by Generalized Sparse PriorConcluding Remarks

20 March, 2013

SPDSA2013, Sendai, Japan

Slide33

Probabilistic Model and Belief Propagation

Probabilistic Information Processing

Probabilistic Models

Bayes Formulas

Belief Propagation

=Bethe Approximation

Bayesian

Networks

Markov Random Fields

20 March, 2013

SPDSA2013, Sendai, Japan

V

: Set of all the nodes (vertices) in graph

G

E

: Set of all the links (edges) in graph

G

j

i

Message

=Effective Field

Slide44

Supervised Learning of Pairwise Markov Random Fields by Loopy Belief

Propagation

Prior

Probability

of natural images is assumed to be the

following pairwise Markov random fields:

4

20 March, 2013

SPDSA2013, Sendai, Japan

Slide5Supervised Learning Scheme by Loopy Belief Propagation in Pairwise Markov random fields:

5

Supervised Learning of Pairwise Markov Random Fields by Loopy Belief Propagation

5

Histogram from

Supervised Data

20 March, 2013

SPDSA2013, Sendai, Japan

M

. Yasuda, S.

Kataoka

and

K.Tanaka

, J. Phys. Soc.

Jpn

, Vol.81, No.4, Article

No.044801,

2012.

Slide66

Supervised Learning of Pairwise Markov Random Fields by Loopy Belief Propagation

Supervised Learning Scheme by Loopy Belief Propagation in Pairwise Markov random fields:

6

20 March, 2013

SPDSA2013, Sendai, Japan

Slide77

Bayesian Image Modeling by Generalized Sparse

Prior

Assumption

: Prior Probability is given as the following Gibbs distribution with the interaction

a

between every nearest neighbour pair of pixels:

7

p

=0:

q-state Potts modelp=2: Discrete Gaussian Graphical Model

20 March, 2013

SPDSA2013, Sendai, Japan

Slide88

Prior in Bayesian Image Modeling

8

In the region of 0<

p

<0.3504…, the first order phase transition appears and the solution

a

*

does not exist.

20 March, 2013

SPDSA2013, Sendai, Japan

Loopy Belief Propagation

q

=16

Slide99

Bayesian Image Modeling by Generalized Sparse Prior: Conditional Maximization of Entropy

9

Lagrange Multiplier

Loopy Belief Propagation

20 March, 2013

SPDSA2013, Sendai, Japan

K. Tanaka

,

M.

Yasuda and D. M.

Titterington

:

J. Phys. Soc.

Jpn

,

81

,

114802

, 2012

.

Slide1010

Prior Analysis by LBP and Conditional Maximization of Entropy in Bayesian Image Modeling

10

Prior Probability

q

=16

p

=0.2

q

=16

p

=0.5

Repeat until

A

converges

20 March, 2013

SPDSA2013, Sendai, Japan

LBP

Slide1111

Prior Analysis by LBP and Conditional Maximization of Entropy in Generalized Sparse Prior

11

Log-Likelihood for

p

when the original image

f

*

is given

q

=16

LBP

Free Energy of Prior

20 March, 2013

SPDSA2013, Sendai, Japan

K. Tanaka

,

M.

Yasuda and D. M.

Titterington

:

J. Phys. Soc.

Jpn

,

81

,

114802

, 2012

.

Slide1212

Degradation Process in Bayesian Image Modeling

Assumption

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

12

20 March, 2013

SPDSA2013, Sendai, Japan

Slide1313

Noise Reductions by Generalized Sparse Prior

Posterior Probability

13

Lagrange Multipliers

20 March, 2013

SPDSA2013, Sendai, Japan

K. Tanaka

,

M.

Yasuda and D. M.

Titterington

:

J. Phys. Soc.

Jpn

,

81

,

114802

, 2012

.

Slide1414

Noise Reduction Procedures Based on LBP and Conditional Maximization of Entropy

14

Repeat until

C

converge

Repeat

until

u

,

B

and

L converge

20 March, 2013

SPDSA2013, Sendai, Japan

Marginals and Free Energy in LBP

Repeat until

C

and

M

converge

Input

p

and data

g

Slide1515

Noise Reductions by Generalized Sparse Priors and Loopy Belief Propagation

p

=0.5

p=0.2

Original Image

Degraded Image

Restored

Image

p=1

20 March, 2013

SPDSA2013, Sendai, Japan

K. Tanaka

,

M.

Yasuda and

D

. M.

Titterington

:

J. Phys. Soc.

Jpn

,

81

,

114802

, 2012

.

Slide1616

Noise Reductions by Generalized Sparse Priors and Loopy Belief Propagation

p

=0.5

p=0.2

Original Image

Degraded Image

Restored

Image

p

=1

20 March, 2013

SPDSA2013, Sendai, Japan

K. Tanaka

,

M.

Yasuda and

D

. M.

Titterington

:

J. Phys. Soc.

Jpn

,

81

,

114802

, 2012

.

Slide1717

Summary

Formulation of Bayesian image modeling for image processing by means of generalized sparse priors and loopy belief propagation are proposed.Our formulation is based on the conditional maximization of entropy with some constraints.In our sparse priors, although the first order phase transitions often appear, our algorithm works well also in such cases.

20 March, 2013

SPDSA2013, Sendai, Japan

Slide1818

References

S. Kataoka, M. Yasuda, K. Tanaka and D. M. Titterington: Statistical Analysis of the Expectation-Maximization Algorithm with Loopy Belief Propagation in Bayesian Image Modeling, Philosophical Magazine: The Study of Condensed Matter, Vol.92, Nos.1-3, pp.50-63,2012.M. Yasuda and K. Tanaka: TAP Equation for Nonnegative Boltzmann Machine: Philosophical Magazine: The Study of Condensed Matter, Vol.92, Nos.1-3, pp.192-209, 2012. S. Kataoka, M. Yasuda and K. Tanaka: Statistical Analysis of Gaussian Image Inpainting Problems, Journal of the Physical Society of Japan, Vol.81, No.2, Article No.025001, 2012. M. Yasuda, S. Kataoka and K.Tanaka: Inverse Problem in Pairwise Markov Random Fields using Loopy Belief Propagation, Journal of the Physical Society of Japan, Vol.81, No.4, Article No.044801, pp.1-8, 2012.M. Yasuda, Y. Kabashima and K. Tanaka: Replica Plefka Expansion of Ising systems, Journal of Statistical Mechanics: Theory and Experiment, Vol.2012, No.4, Article No.P04002, pp.1-16, 2012.K. Tanaka, M. Yasuda and D. M. Titterington: Bayesian image modeling by means of generalized sparse prior and loopy belief propagation, Journal of the Physical Society of Japan, Vol.81, No.11, Article No.114802, 2012.M. Yasuda and K. Tanaka: Susceptibility Propagation by Using Diagonal Consistency, Physical Review E, Vol.87, No.1, Article No.012134, 2013.

20 March, 2013

SPDSA2013, Sendai, Japan

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