1. ICPR2004 (24 July, 2004, Cambridge) 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).

2. ICPR2004 (24 July, 2004, Cambridge) 2 1. Introduction Probabilistic Image Processing Image Processing Massive Probabilistic Model Computational Complexity Loopy Belief Propagation (LBP) Approximate 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). Practical Algorithm

3. ICPR2004 (24 July, 2004, Cambridge) 3 Standard Regularization and Probabilistic Image Processing Cost Function of Standard Regularization 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. Original Image Degraded Image

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

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

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

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

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

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

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

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

12. ICPR2004 (24 July, 2004, Cambridge)12 4. Numerical Experiments Hyperparameters are determined so as to maximize the marginal likelihood. Degraded Image LBP Original Image Mean Field Method Mean-Field Method (MF)Loopy Belief Propagation (LBP) [0.4600,0.4618][0.6193,0.6219] [0.2163,0.2177][0.2689,0.2704] [2.2736,2.8054][4.5860,4.6509] Confidence intervals for 20 samples of original images

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

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

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

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

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

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

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

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

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

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

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