Loopy belief propagation and probabilistic image processing K

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

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. Similarity Ordered states are determined from interactions and external fields. Para It is difficult for conventional filters to treat fluctuation in data. Critical Ferro Fluctuation is enhanced near critical temperature. 2

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

Probabilistic Image Processing Original Image Degraded Image Bayes Formula 4

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

Bayes Formula and A Posteriori Probability 6

Maximization of Posterior Marginal 7

Hyperparameter Estimation Maximization of Marginal Likelihood Marginalize 8

Deterministic Equation of Loopy Belief Propagation 9

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

Approximate Normalization in Loopy Belief Propagation 11

Binary Image Restoration Original images are generated by Monte Carlo simulations in the a priori probability. Original Image Degraded Image (p=0. 2) Restored Image 12

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

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

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 Q-state Potts Model 15

Multi-Valued Image Restoration (Q=4) Hyperparameters are determined so as to maximize the marginal likelihood. Original Image Degraded Image （3 p=0. 3） Q-state Potts Model Q-Ising Model 16

Gray-Level Image Restoration Original Image Belief Propagation Lowpass Filter MSE: 2075 MSE: 244 MSE: 217 MSE: 3469 MSE: 371 Degraded Image MSE: 523 Median Filter MSE: 135 MSE: 395 17

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 18

Appendix A: Bethe Free Energy 19

Appendix A: Basic Framework of Bethe Approximation Constraint Conditions 20

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

Appendix B: Original images are generated by Monte Carlo simulations in the a priori probability. Degraded Image Mean Field Original Image Approx. (p=0. 2) Loopy Belief Propagation 22

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

Appendix C: 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 24

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