Cluster Variation Method and Probabilistic Image Processing -- Loopy Belief Propagation --

1. Cluster Variation Method and Probabilistic Image Processing-- Loopy Belief Propagation -- Kazuyuki Tanaka Graduate School of Information Sciences Tohoku University kazu@statp.is.tohoku.ac.jp http://www.statp.is.tohoku.ac.jp/~kazu/index-e.html University of Glasgow

2. Noise Probabilistic Model and Image Restoration Transmission Original Image Degraded Image University of Glasgow

3. Image Restoration and Magnetic Material Regular lattice consisting of a lot of nodes. Neighbouring spin interactions and Markov random field Restored images are determined from a priori information and given data. Ordered states are determined from interactions and external fields. Interpretation and prediction of physical property by means of physical model. Feature detection from the data and image processing by means of filters. Similarity of Mathematical Structure University of Glasgow

4. Massive Information Processing and Probabilistic Information Processing • Computational Complexity. • Approximation algorithms for massive information processing by means of advanced mean-field methods. Application of the cluster variation method (Bethe/Kikuchi method) to massive information processing Cluster Variation Method is equivalent to a generalized loopy belief propagation for probabilistic inference in the artificial intelligence. University of Glasgow

5. Important point in the application of cluster variation method to probabilistic image processing • Design of iterative algorithms for probabilistic inference based on cluster variation method (Computer Science). • Hyperparameter estimation (Statistics). • Cooperative phenomena in probabilistic models and probabilistic information processing (Physics). University of Glasgow

6. A Priori Probability Degradation Process (Binary Symmetric Channel) Degradation Process and A Priori Probability in Binary Image Restoration University of Glasgow

7. A Priori Probability in Binary Image Restoration University of Glasgow

8. Bayes Formula and A Posteriori Probability Maximization of Posterior Marginal University of Glasgow

9. A Posteriori Probability in Binary Image Restoration University of Glasgow

10. Kullback-Leibler divergence University of Glasgow

11. Kullback-Leibler Divergence University of Glasgow

12. Basic Framework of Pair Approximation in Cluster Variation Method University of Glasgow

13. Update Rule is reduced to Loopy Belief Propagation Propagation Rule of Pair Approximation in Cluster Variation Method University of Glasgow

14. One-Body Marginal Probability of Pair Approximation in CVM University of Glasgow

15. Two-Body Marginal Probability of Pair Approximation in CVM University of Glasgow

16. Message Propagation Rule of Pair Approximation in CVM University of Glasgow

17. Degraded Image (p=0.2) Original Image Restored Image Binary Image Restoration Original images are generated by Monte Carlo simulations in the a priori probability. University of Glasgow

18. Binary Image Restoration Original Image Degraded Image Restored Image University of Glasgow

19. Maximization of Marginal Likelihood Hyperparameter Estimation University of Glasgow

20. Binary Image Restoration Degraded Image(p=0.2) Original images are generated by Monte Carlo simulations in the a priori probability. Original Image Restored Image University of Glasgow

21. Hyperparameters are determined so as to maximize the marginal likelihood. Binary Image Restoration Mean Field Approximation Pair Approximation in CVM Original Image Degraded Image University of Glasgow

22. Multi-Valued Image Restoration Degradation Process University of Glasgow

23. A Priori Probability in Multi-Valued Image Restoration Q-state Ising Model Q-state Potts Model Kronecker Delta University of Glasgow

24. Multi-Valued Image Restoration (Q=4) Hyperparameters are determined so as to maximize the marginal likelihood. 4-state Ising Model Degraded Image Original Image Restored Image 4-state Potts Model University of Glasgow

25. Multi-Valued Image Restoration (Q=4) Hyperparameters are determined so as to maximize the marginal likelihood. 4-state Potts Model 4-state Ising Model Degraded Image（3p=0.3） Original Image University of Glasgow

26. Summary • Probabilistic Image Processing by Bayes Formula • Cluster Variation Method and Loopy Belief Propagation • Binary Image Restoration • Multi-Valued Image Restoration University of Glasgow

27. Other Practical Applications • Edge Detection • Segmentation • Texture Analysis • Image Compression • Motion Detection • Color Image University of Glasgow

28. Other Theoretical Works • Hyperparameter Estimation by EM algorithm • Statistical Performance Estimation and Spin Glass Theory • Replica method • Inequality of Statistical Quantity • Line Field • Generalized Loopy Belief Propagation and Cluster Variation Method • Information Geometry and Cluster Variation Method University of Glasgow