Loopy belief propagation and probabilistic image processing

1. Loopy belief propagation andprobabilistic image processing K. Tanaka (Tohoku University, Japan) J. Inoue (Hokkaido University, Japan) D. M. Titterington (University of Glasgow, UK)

2. Image Processing and Magnetic Material Regular lattice consisting of a lot of nodes. Interactions among neighboring nodes Output images are determined from a priori information and given data. Ordered states are determined from interactions and external fields. Similarity Para Critical Ferro It is difficult for conventional filters to treat fluctuation in data. Fluctuation is enhanced near critical temperature.

3. Noise Probabilistic Model and Image Restoration Transmission Original Image Degraded Image

4. Probabilistic Image Processing Degraded Image Original Image Bayes Formula

5. Degradation Process (Binary Symmetric Channel) A Priori Probability Degradation Process and A Priori Probability in Binary Image Restoration

6. Bayes Formula and A Posteriori Probability

7. Maximization of Posterior Marginal Maximization of Posterior Marginal

8. Marginalize Hyperparameter Estimation Maximization of Marginal Likelihood

9. Deterministic Equation of Loopy Belief Propagation

10. Message Update Rule of Loopy Belief Propagation Fixed-Point Equations Natural Iteration

11. Approximate Normalization in Loopy Belief Propagation

12. 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

13. Hyperparameters are determined so as to maximize the marginal likelihood. Binary Image Restoration Loopy Belief Propagation Original Image Degraded Image

14. A Priori Probability in Multi-Valued Image Restoration Q-Ising Model Q-state Potts Model

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

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

17. Gray-Level Image Restoration 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

18. 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

19. Appendix A: Bethe Free Energy

20. Appendix A: Basic Framework of Bethe Approximation Constraint Conditions

21. Update Rule is reduced to Loopy Belief Propagation Appendix A: Propagation Rule of Bethe Approximation

22. Appendix B: Original images are generated by Monte Carlo simulations in the a priori probability. Degraded Image(p=0.2) Loopy Belief Propagation Mean Field Approx. Original Image

23. Appendix B: Hyperparameters are determined so as to maximize the marginal likelihood. Mean-Field Approx. Loopy Belief Propagation Original Image Degraded Image

24. Appendix C: Multi-Valued Image Restoration (Q=4) Hyperparameters are determined so as to maximize the marginal likelihood. Q-Ising Model Original Image Restored Image Degraded Image

25. Appendix C: Multi-Valued Image Restoration (Q=4) Hyperparameters are determined so as to maximize the marginal likelihood. Q-state Potts Model Restored Image Original Image Degraded Image