1. Efficient Belief Propagation for Image Restoration Qi Zhao Mar.22,2006

2. References Pedro F. Felzenszwalb and Daniel P. Huttenlocher. Efficient Belief Propagation for Early Vision. To appear in the International Journal of Computer Vision. Y.Weiss and W.T. Freeman. On the optimality of solutions of themax-product belief propagation algorithm in arbitrary graphs. IEEE Transactions on Information Theory, 47(2):723–735, 2001. L.I. Rudin, S. Osher, and E. Fatemi, "NONLINEAR TOTAL VARIATION BASED NOISE REMOVAL ALGORITHMS", PHYSICA D 60 (1-4): 259-268 Nov. 1, 1992.

3. Outline MRF model for Image Restoration Image Restoration using Efficient Belief Propagation Experimental Results(Demo) –Additive noise removal –Image Inpaiting

4. Markov Model Motivation –Markov random field models provide a robust and unified framework for early vision problems.

5. MRF Models for Image Restoration : set of pixels in an image : a finite set of labels, which correspond to the underlying intensities of the pixels. E.g.,, where Objective: Finding a labeling that minimizes the energy corresponds to the MAP estimation problem for the defined MRF. –Neighborhood system: 4-neighborhood system –Prior: pair-wise potential cliques –Likelihood energy: e.g.,where –Posterior energy:

6. Loopy Belief Propagation: Max-Product Let be the message that node sends to a neighboring node at iteration, we have Finally, the label that maximizes is individually selected for each node.

7. Speed Up Techniques (1) Computing a Message Update in Linear Time –Computing, where Potts Model: Firstly, compute the lower envelope of the parabolas; Secondly, fill in the value of by checking the height of the lower envelope at each grid location.

8. Speed Up Techniques (2) BP on the Grid Graph –The grid graph is bipartite –Two groups of nodes: A & B –Time t: Msg from Nodes A -> Nodes B –Time t+1: Msg from Nodes B -> Nodes A

9. Speed Up Techniques (3) Multi-Grid BP –Problem in BP: it takes many iterations for information to flow over large distances in the grid graph. –Basic Idea: to perform BP in a coarse-to-fine manner, so that long range interactions between pixels can be captured by short paths in coarse graphs.

10. Experiments (1) Noise Removal Original Image BP Restored Image

11. Parameters Experiments (2) Original Image

12. Experiments (3) Comparisons TV Restored Image BP Restored Image

13. Experiments (4) Image Impainting –For pixel in the masked region, (a) Noised, masked image (b) L=1,T=25 (c) L=1,T=14 (d) L=5,T=5 Efficiency Improved by the Coarse-to-Fine Technique!

14. Thank You!