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Probabilistic image processing based on the Q -Ising model by means of the mean-field method and loopy belief propagati

Probabilistic image processing based on the Q -Ising model by means of the mean-field method and loopy belief propagation. Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University, Japan kazu@statp.is.tohoku.ac.jp http://www.statp.is.tohoku.ac.jp/~kazu/ and

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Probabilistic image processing based on the Q -Ising model by means of the mean-field method and loopy belief propagati

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  1. Probabilistic image processing based on the Q-Ising model by means of the mean-field method and loopy belief propagation Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University, Japan kazu@statp.is.tohoku.ac.jp http://www.statp.is.tohoku.ac.jp/~kazu/ and D. M. Titterington Department of Statistics, University of Glasgow, UK References: K. Tanaka: J. Phys. A, 35, R81 （2002). J. Inoue and K. Tanaka: J. Phys. A, 36, 10997 (2003). K. Tanaka, J. Inoue and D. M. Titterington: J. Phys. A, 36, 11023 (2003). ICPR2004 (24 July, 2004, Cambridge)

  2. Probabilistic Image Processing 1. Introduction Image Processing Massive Probabilistic Model Computational Complexity Loopy Belief Propagation (LBP) Approximate Algorithm Practical Algorithm • Probabilistic model with tree graphical structure: • Belief Propagation => Exact • Probabilistic model with loopy graphical structure: • Loopy Belief propagation => Approximation • A practical algorithm for image restoration based on loopy belief propagation (LBP). ICPR2004 (24 July, 2004, Cambridge)

  3. Cost Function of Standard Regularization Standard Regularization and Probabilistic Image Processing Original Image Degraded Image • Corresponding Posterior in Bayes Statistics Similarity to Q-Ising model or Gaussian Graphical model • Efficiency of Q-Ising model for probabilistic image processing, particularly, hyperparameter estimation. ICPR2004 (24 July, 2004, Cambridge)

  4. 2. Probabilistic Image Processing Degraded Image Original Image Bayes Formula ICPR2004 (24 July, 2004, Cambridge)

  5. A Priori Probability in Multi-Valued Image Restoration Similarity ICPR2004 (24 July, 2004, Cambridge)

  6. Degradation Process in Multi-Valued Image Restoration Degraded Image Original Image ICPR2004 (24 July, 2004, Cambridge)

  7. Bayes Formula and A Posteriori Probability ICPR2004 (24 July, 2004, Cambridge)

  8. Marginalize Hyperparameter Estimation Maximization of Marginal Likelihood ICPR2004 (24 July, 2004, Cambridge)

  9. 3. Loopy Belief Propagation Maximization of Posterior Marginal ICPR2004 (24 July, 2004, Cambridge)

  10. Deterministic Equation of Loopy Belief Propagation ICPR2004 (24 July, 2004, Cambridge)

  11. Message Update Rule of Loopy Belief Propagation Fixed-Point Equations Natural Iteration ICPR2004 (24 July, 2004, Cambridge)

  12. 4. Numerical Experiments Mean Field Method LBP Hyperparameters are determined so as to maximize the marginal likelihood. Original Image Degraded Image Confidence intervals for 20 samples of original images ICPR2004 (24 July, 2004, Cambridge)

  13. Multi-Valued Image Restoration (Q=4) Hyperparameters are determined so as to maximize the marginal likelihood. Loopy Belief Propagation Degraded Image（(Q-1)p=0.3） Mean Field Method Original Image ICPR2004 (24 July, 2004, Cambridge)

  14. Image Restoration by means of Gaussian Graphical Model and Loopy Belief Propagation Additive White Gaussian Noise Loopy Belief Propagation Original Image Degraded Image Mean Field Method MSE: 591 MSE: 325 MSE: 1512 Exact Calculation Lowpass Filter Median Filter Wiener Filter MSE:315 MSE: 411 MSE: 545 MSE: 447 ICPR2004 (24 July, 2004, Cambridge)

  15. Comparison with Standard Regularization and Constrained Least Mean Square Filter Additive White Gaussian Noise (σ=40) Constrained Least Mean Square Filter Loopy Belief Propagation Original Image Degraded Image MSE: 325 MSE: 1512 MSE: 372 ICPR2004 (24 July, 2004, Cambridge)

  16. 5. Summary • Probabilistic Image Processing by Bayes Formula and Loopy Belief Propagation • Some Numerical Experiments Future Problems • Segmentation • Image Compression • Motion Detection • Color Image • EM algorithm • Statistical Performance • Line Fields ICPR2004 (24 July, 2004, Cambridge)

  17. Appendix A: Graphical Probabilistic Model ICPR2004 (24 July, 2004, Cambridge)

  18. Appendix A: Kullback-Leibler divergence ICPR2004 (24 July, 2004, Cambridge)

  19. Appendix A: Bethe Free Energy ICPR2004 (24 July, 2004, Cambridge)

  20. Appendix A: Basic Framework of Bethe Approximation ICPR2004 (24 July, 2004, Cambridge)

  21. Update Rule is reduced to Loopy Belief Propagation Appendix A: Propagation Rule of Bethe Approximation ICPR2004 (24 July, 2004, Cambridge)

  22. Appendix B: Multi-Valued Image Restoration (Q=4) Hyperparameters are determined so as to maximize the marginal likelihood. Q-Ising Model Original Image Degraded Image Restored Image ICPR2004 (24 July, 2004, Cambridge)

  23. In the adaptive Wiener filter, the assumption is assumed and the variation is estimated from the given degraded image. Appendix C: Wiener Filter ICPR2004 (24 July, 2004, Cambridge)

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