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. 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)

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Statistical performance analysis by loopy belief propagation in probabilistic image processing

Statistical performance analysis by loopy belief propagation in probabilistic image processing

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
Introduction in

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 by Bayesian Network in

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
MRF, Belief Propagation and Statistical Performance in

  • 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
    Outline in

    • 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)


    Outline1
    Outline in

    • 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)


    Image restoration by bayesian statistics

    Image Restoration in by Bayesian Statistics

    Original Image

    LRI Seminar 2010 (Univ. Paris-Sud)


    Image restoration by bayesian statistics1

    Image Restoration in by Bayesian Statistics

    Noise

    Transmission

    Original Image

    Degraded Image

    LRI Seminar 2010 (Univ. Paris-Sud)


    Image restoration by bayesian statistics2

    Image Restoration in by Bayesian Statistics

    Noise

    Transmission

    Original Image

    Degraded Image

    Estimate

    LRI Seminar 2010 (Univ. Paris-Sud)


    Image restoration by bayesian statistics3

    Image Restoration in by Bayesian Statistics

    Noise

    Transmission

    Posterior

    Original Image

    Degraded Image

    Estimate

    Bayes Formula

    10

    LRI Seminar 2010 (Univ. Paris-Sud)


    Image restoration by bayesian statistics4

    Image Restoration in by Bayesian Statistics

    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)


    Image restoration by bayesian statistics5

    Image Restoration in by Bayesian Statistics

    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


    Bayesian image analysis

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

    Bayesian Image Analysis

    Prior Probability

    LRI Seminar 2010 (Univ. Paris-Sud)


    Bayesian image analysis1

    Bayesian Image Analysis in

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

    Prior Probability

    14

    LRI Seminar 2010 (Univ. Paris-Sud)


    Bayesian image analysis2

    Bayesian Image Analysis in

    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 analysis3
    Bayesian Image Analysis in

    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 analysis4
    Bayesian Image Analysis in

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

    LRI Seminar 2010 (Univ. Paris-Sud)

    17


    Bayesian image analysis5
    Bayesian Image Analysis in

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

    LRI Seminar 2010 (Univ. Paris-Sud)

    18


    Bayesian image analysis6
    Bayesian Image Analysis in

    Degraded

    Image

    Original

    Image

    Degradation Process

    Prior Probability

    Posterior Probability

    Estimate

    LRI Seminar 2010 (Univ. Paris-Sud)


    Bayesian image analysis7
    Bayesian Image Analysis in

    Degraded

    Image

    Original

    Image

    Degradation Process

    Prior Probability

    Posterior Probability

    Estimate

    Data Dominant

    Smoothing

    LRI Seminar 2010 (Univ. Paris-Sud)

    20


    Bayesian image analysis8
    Bayesian Image Analysis in

    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 analysis9
    Bayesian Image Analysis in

    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
    Image Restorations by in 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)


    Outline2
    Outline in

    • 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
    Statistical Performance in by Sample Average of Numerical Experiments

    Original Images

    LRI Seminar 2010 (Univ. Paris-Sud)


    Statistical performance by sample average of numerical experiments1
    Statistical Performance in 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 experiments2
    Statistical Performance in 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 experiments3
    Statistical Performance in 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
    Statistical Performance Estimation in

    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
    Statistical Performance Estimation for in Gauss Markov Random Fields

    s=40

    s=40

    a

    a

    LRI Seminar 2010 (Univ. Paris-Sud)


    Outline3
    Outline in

    • 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
    Marginal Probability in Belief Propagation in

    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)


    Marginal probability in belief propagation1

    2 in

    Marginal Probability in Belief Propagation

    LRI Seminar 2010 (Univ. Paris-Sud)


    Marginal probability in belief propagation2

    2 in

    2

    Marginal Probability in Belief Propagation

    In 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)


    Marginal probability in belief propagation3

    1 in

    2

    Marginal Probability in Belief Propagation

    In 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)


    Marginal probability in belief propagation4

    1 in

    2

    1

    2

    Marginal Probability in Belief Propagation

    In 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
    Belief Propagation in in Probabilistic Image Processing

    LRI Seminar 2010 (Univ. Paris-Sud)


    Belief propagation in probabilistic image processing1
    Belief Propagation in in Probabilistic Image Processing

    LRI Seminar 2010 (Univ. Paris-Sud)


    Image restorations by gaussian markov random fields and conventional filters1
    Image Restorations by in 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
    Gray-Level Image Restoration in (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)


    Binary image restoration by loopy belief propagation
    Binary Image Restoration by in Loopy Belief Propagation

    a*=0.465

    LRI Seminar 2010 (Univ. Paris-Sud)


    Statistical performance by sample average of numerical experiments4
    Statistical Performance in 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 fields

    = in

    >

    =

    Statistical Performance Estimation for Binary Markov Random Fields

    Light 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 fields1
    Statistical Performance Estimation for Binary Markov Random Fields

    LRI Seminar 2010 (Univ. Paris-Sud)


    Statistical performance estimation for markov random fields
    Statistical Performance Estimation for Markov Random Fields 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 fields1
    Statistical Performance Estimation for Markov Random Fields Fields

    LRI Seminar 2010 (Univ. Paris-Sud)


    Outline4
    Outline Fields

    • 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
    Summary Fields

    • 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
    Image Impainting by Gauss MRF and LBP Fields

    Our framework can be extended to erase a scribbling.

    Gauss MRF

    and LBP

    LRI Seminar 2010 (Univ. Paris-Sud)


    References
    References Fields

    • 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)