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Statistical performance analysis by loopy belief propagation in probabilistic image processing

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### Statistical performance analysis by loopy belief propagation in probabilistic image processing

### Image Restoration by Bayesian Statistics

### Image Restoration by Bayesian Statistics

### Image Restoration by Bayesian Statistics

### Image Restoration by Bayesian Statistics

### Image Restoration by Bayesian Statistics

### Bayesian Image Analysis

### Bayesian Image Analysis

### Bayesian Image Analysis

OutlineOutline

Kazuyuki Tanaka

Graduate School of Information Sciences,

Tohoku University, Japan

http://www.smapip.is.tohoku.ac.jp/~kazu/

Collaborators

D. M. Titterington (University of Glasgow, UK)

M. Yasuda (Tohoku University, Japan)

S. Kataoka (Tohoku University, Japan)

LRI Seminar 2010 (Univ. Paris-Sud)

Introduction

Bayesian network is originally one of the methods for probabilistic inferences in artificial intelligence.

Some probabilistic models for information processing are also regarded as Bayesian networks.

Bayesian networks are expressed in terms of products of functions with a couple of random variables and can be associated with graphical representations.

Such probabilistic models for Bayesian networks are referred to as Graphical Model.

LRI Seminar 2010 (Univ. Paris-Sud)

Probabilistic Image Processing by Bayesian Network

Probabilistic image processing systems are formulated on square grid graphs.

Averages, variances and covariances of the Bayesian network are approximately computed by using the belief propagation on the square grid graph.

LRI Seminar 2010 (Univ. Paris-Sud)

MRF, Belief Propagation and Statistical Performance

- Geman and Geman (1986): IEEE Transactions on PAMI
- Image Processing by Markov Random Fields (MRF)

- Tanaka and Morita (1995): Physics Letters A
- Cluster Variation Method for MRF in Image Processing
- CVM= Generalized Belief Propagation (GBP)

- Nishimori and Wong (1999): Physical Review E
- Statistical Performance Estimation for MRF
- (Infinite Range Ising Model and Replica Theory)

Is it possible to estimate the performance of belief propagation statistically?

IW-SMI2010 (Kyoto)

Outline

- Introduction
- Bayesian Image Analysis by Gauss Markov Random Fields
- Statistical Performance Analysis for Gauss Markov Random Fields
- Statistical Performance Analysis in Binary Markov Random Fields by Loopy Belief Propagation
- Concluding Remarks

LRI Seminar 2010 (Univ. Paris-Sud)

Outline

- Introduction
- Bayesian Image Analysis by Gauss Markov Random Fields
- Statistical Performance Analysis for Gauss Markov Random Fields
- Statistical Performance Analysis in Binary Markov Random Fields by Loopy Belief Propagation
- Concluding Remarks

LRI Seminar 2010 (Univ. Paris-Sud)

Noise

Transmission

Original Image

Degraded Image

LRI Seminar 2010 (Univ. Paris-Sud)

Noise

Transmission

Original Image

Degraded Image

Estimate

LRI Seminar 2010 (Univ. Paris-Sud)

Noise

Transmission

Posterior

Original Image

Degraded Image

Estimate

Bayes Formula

10

LRI Seminar 2010 (Univ. Paris-Sud)

Noise

Transmission

Posterior

Original Image

Degraded Image

Estimate

Assumption 1: Original images are randomly generated by according to a prior probability.

Bayes Formula

11

LRI Seminar 2010 (Univ. Paris-Sud)

Noise

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

Transmission

Posterior

Original Image

Degraded Image

Estimate

Bayes Formula

12

LRI Seminar 2010 (Univ. Paris-Sud)

7July, 2010

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

Prior Probability

LRI Seminar 2010 (Univ. Paris-Sud)

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

Prior Probability

14

LRI Seminar 2010 (Univ. Paris-Sud)

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

Prior Probability

Patterns by MCMC.

15

LRI Seminar 2010 (Univ. Paris-Sud)

Bayesian Image Analysis

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

V:Set of all the pixels

LRI Seminar 2010 (Univ. Paris-Sud)

Bayesian Image Analysis

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

LRI Seminar 2010 (Univ. Paris-Sud)

17

Bayesian Image Analysis

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

LRI Seminar 2010 (Univ. Paris-Sud)

18

Bayesian Image Analysis

Degraded

Image

Original

Image

Degradation Process

Prior Probability

Posterior Probability

Estimate

LRI Seminar 2010 (Univ. Paris-Sud)

Bayesian Image Analysis

Degraded

Image

Original

Image

Degradation Process

Prior Probability

Posterior Probability

Estimate

Data Dominant

Smoothing

LRI Seminar 2010 (Univ. Paris-Sud)

20

Bayesian Image Analysis

Degraded

Image

Original

Image

Degradation Process

Prior Probability

Posterior Probability

Estimate

Bayesian Network

Data Dominant

Smoothing

11 March, 2010

IW-SMI2010 (Kyoto)

LRI Seminar 2010 (Univ. Paris-Sud)

21

Bayesian Image Analysis

Degraded

Image

Original

Image

Degradation Process

Prior Probability

Posterior Probability

Estimate

Bayesian Network

Data Dominant

Smoothing

LRI Seminar 2010 (Univ. Paris-Sud)

22

Image Restorations by Gaussian Markov Random Fields and Conventional Filters

Original Image

Degraded Image

Restored

Image

V: Set of all

the pixels

Gauss Markov Random Field

(3x3) Lowpass

(5x5) Median

LRI Seminar 2010 (Univ. Paris-Sud)

Outline

- Introduction
- Bayesian Image Analysis by Gauss Markov Random Fields
- Statistical Performance Analysis for Gauss Markov Random Fields
- Statistical Performance Analysis in Binary Markov Random Fields by Loopy Belief Propagation
- Concluding Remarks

LRI Seminar 2010 (Univ. Paris-Sud)

Statistical Performance by Sample Average of Numerical Experiments

Original Images

LRI Seminar 2010 (Univ. Paris-Sud)

Statistical Performance by Sample Average of Numerical Experiments

Noise Pr{G|F=f,s}

Original Images

Observed

Data

LRI Seminar 2010 (Univ. Paris-Sud)

26

Statistical Performance by Sample Average of Numerical Experiments

Noise Pr{G|F=f,s}

Posterior Probability

Pr{F|G=g,a,s}

Original Images

Observed

Data

Estimated Results

11 March, 2010

IW-SMI2010 (Kyoto)

LRI Seminar 2010 (Univ. Paris-Sud)

27

Statistical Performance by Sample Average of Numerical Experiments

Sample Average of Mean Square Error

Noise Pr{G|F=f,s}

Posterior Probability

Pr{F|G=g,a,s}

Original Images

Observed

Data

Estimated Results

LRI Seminar 2010 (Univ. Paris-Sud)

28

Statistical Performance Estimation

Original Image

Degraded Image

Additive White Gaussian Noise

Restored

Image

Posterior Probability

Additive White Gaussian Noise

LRI Seminar 2010 (Univ. Paris-Sud)

Statistical Performance Estimation for Gauss Markov Random Fields

s=40

s=40

a

a

LRI Seminar 2010 (Univ. Paris-Sud)

- Introduction
- Bayesian Image Analysis by Gauss Markov Random Fields
- Statistical Performance Analysis for Gauss Markov Random Fields
- Statistical Performance Analysis in Binary Markov Random Fields by Loopy Belief Propagation
- Concluding Remarks

LRI Seminar 2010 (Univ. Paris-Sud)

Marginal Probability in Belief Propagation

In order to compute the marginal probability Pr{F2|G=g},

we take summations over all the pixels except the pixel 2.

LRI Seminar 2010 (Univ. Paris-Sud)

2

Marginal Probability in Belief PropagationIn the belief propagation, the marginal probability Pr{F2|G=g} is approximately expressed in terms of the messages from the neighbouring region of the pixel 2.

LRI Seminar 2010 (Univ. Paris-Sud)

2

Marginal Probability in Belief PropagationIn order to compute the marginal probability Pr{F1,F2|G=g},

we take summations over all the pixels except the pixels 1 and2.

LRI Seminar 2010 (Univ. Paris-Sud)

2

1

2

Marginal Probability in Belief PropagationIn the belief propagation, the marginal probability Pr{F1,F2|G=g} is approximately expressed in terms of the messages from the neighbouring region of the pixels 1 and 2.

LRI Seminar 2010 (Univ. Paris-Sud)

Belief Propagation in Probabilistic Image Processing

LRI Seminar 2010 (Univ. Paris-Sud)

Belief Propagation in Probabilistic Image Processing

LRI Seminar 2010 (Univ. Paris-Sud)

Image Restorations by Gaussian Markov Random Fields and Conventional Filters

Original Image

Degraded Image

Restored

Image

V: Set of all

the pixels

Belief Propagation

Exact

LRI Seminar 2010 (Univ. Paris-Sud)

Gray-Level Image Restoration(Spike Noise)

Original Image

Belief

Propagation

Degraded

Image

Lowpass Filter

Median Filter

MSE: 244

MSE: 217

MSE:135

MSE: 2075

MSE: 3469

MSE: 371

MSE: 523

MSE: 395

LRI Seminar 2010 (Univ. Paris-Sud)

Statistical Performance by Sample Average of Numerical Experiments

Sample Average of Mean Square Error

Noise Pr{G|F=f,s}

Posterior Probability

Pr{F|G=g,a,s}

Original Images

Observed

Data

Estimated Results

LRI Seminar 2010 (Univ. Paris-Sud)

42

>

=

Statistical Performance Estimation for Binary Markov Random FieldsLight intensities of the original image can be regarded as spin states of ferromagnetic system.

=

>

=

Free Energy of Ising Model with Random External Fields

It can be reduced to the calculation of the average of free energy with respect to locally non-uniform external fields g1, g2,…,g|V|.

LRI Seminar 2010 (Univ. Paris-Sud)

Statistical Performance Estimation for Binary Markov Random Fields

LRI Seminar 2010 (Univ. Paris-Sud)

Statistical Performance Estimation for Markov Random Fields

Multi-dimensional Gauss

Integral Formulas

Spin Glass Theory in Statistical Mechanics

Loopy Belief Propagation

0.8

0.6

0.4

s=40

0.2

s=1

a

a

LRI Seminar 2010 (Univ. Paris-Sud)

Statistical Performance Estimation for Markov Random Fields

LRI Seminar 2010 (Univ. Paris-Sud)

- Introduction
- Bayesian Image Analysis by Gauss Markov Random Fields
- Statistical Performance Analysis for Gauss Markov Random Fields
- Statistical Performance Analysis in Binary Markov Random Fields by Loopy Belief Propagation
- Concluding Remarks

LRI Seminar 2010 (Univ. Paris-Sud)

Summary

- Formulation of probabilistic model for image processing by means of conventional statistical schemes has been summarized.
- Statistical performance analysis of probabilistic image processing by using Gauss Markov Random Fields has been shown.
- One of extensions of statistical performance estimation to probabilistic image processing with discretestates has been demonstrated.

LRI Seminar 2010 (Univ. Paris-Sud)

Image Impainting by Gauss MRF and LBP

Our framework can be extended to erase a scribbling.

Gauss MRF

and LBP

LRI Seminar 2010 (Univ. Paris-Sud)

References

- K. Tanaka and D. M. Titterington: Statistical Trajectory of Approximate EM Algorithm for Probabilistic Image Processing, Journal of Physics A: Mathematical and Theoretical, vol.40, no.37, pp.11285-11300, 2007.
- M. Yasuda and K. Tanaka: The Mathematical Structure of the Approximate Linear Response Relation, Journal of Physics A: Mathematical and Theoretical, vol.40, no.33, pp.9993-10007, 2007.
- K. Tanaka and K. Tsuda: A Quantum-Statistical-Mechanical Extension of Gaussian Mixture Model, Journal of Physics: Conference Series, vol.95, article no.012023, pp.1-9, January 2008
- K. Tanaka: Mathematical Structures of Loopy Belief Propagation and Cluster Variation Method, Journal of Physics: Conference Series, vol.143, article no.012023, pp.1-18, 2009
- M. Yasuda and K. Tanaka: Approximate Learning Algorithm in Boltzmann Machines, Neural Computation, vol.21, no.11, pp.3130-3178, 2009.
- S. Kataoka, M. Yasuda and K. Tanaka: Statistical Performance Analysis in Probabilistic Image Processing, Journal of the Physical Society of Japan, vol.79, no.2, article no.025001, 2010.

LRI Seminar 2010 (Univ. Paris-Sud)

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