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Understanding Belief Propagation and its Applications Dan Yuan June 2004.

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1 Understanding Belief Propagation and its Applications Dan Yuan June 2004

2 Outline Motivation Rationale Applications

3 Probabilistic Inference Directed Graph—Bayesian Network Undirected Graph– Markov Random Field NP-hard Problem: Computing the a posteriori beliefs of RVs in both of these graphs

4 Solutions Approximate Inference MCMC Sampling Belief Propagation

5 Parameterization and conditioning in Undirected Graph The Joint Probability where Z is a normalizing constant There is a cost named compatibility on each link between two neighboring nodes. We assume only the pair-wise compatibility between two nodes. P can be thought of as factoring into five multiplicative potential functions : A BC EAEA EBEB ECEC

6 Parameterization and conditioning in Undirected Graph with a Loop Formulation: A B C EAEA EBEB ECEC Why do we care about loopy graphs?

7 Probability Propagation The max-product update where denotes a normalizing constant and means all nodes neighboring except.

8 Probability Propagation (Cont’d) 1. The algorithm converges to a unique fixed belief regardless of initial conditions in a finite number of iterations. 2. At convergence, the belief for any value of a node i is the maximum of the posterior, conditioned on that node having the value: 3. Define the max-product assignment, by (assuming a unique maximizing value exists). Then is the MAP assignment.

9 Relation to Junction Tree Algorithm Transformation from a general graph to a junction tree, and BP on the junction tree is equivalent to that on the original graph. Transformation is too complicated when the original graph is very loopy.

10 Applications of BP in Computer Vision Unwrapping phase images[Frey, NIPS] Stereo matching [Sun,ECCV ] Shape and reflectance inference from photograph [Weiss, ICCV] Image detail extrapolating [Freeman, IJCV]

11 Experiments Noise Removal Image segmentation Enhancement m ii (x i ) yjyj yiyi …… xjxj xixi … … … … …… …

12 Results—noise removal Pepper and saltWhite gaussian

13 Results—Image Segmentation Enhancement

14 Thanks Questions?

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