Understanding Belief Propagation and its Applications

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Understanding Belief Propagation and its
Applications

• Dan Yuan
• June 2004

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Outline

• Motivation
• Rationale
• Applications

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Probabilistic Inference

• Directed GraphBayesian Network
• Undirected Graph Markov Random Field
• NP-hard Problem Computing the a posteriori
  beliefs of RVs in both of these graphs

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Solutions

• Approximate Inference
• MCMC Sampling
• Belief Propagation

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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
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Parameterization and conditioning in Undirected
Graph with a Loop

• Formulation

Why do we care about loopy graphs?
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Probability Propagation

• The max-product update

where denotes a normalizing constant and
means all nodes neighboring except .
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Probability Propagation (Contd)

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

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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.

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Applications of BP in Computer Vision

• Unwrapping phase imagesFrey, NIPS
• Stereo matching Sun,ECCV
• Shape and reflectance inference from photograph
  Weiss, ICCV
• Image detail extrapolating Freeman, IJCV

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Experiments

• Noise Removal
• Image segmentation Enhancement

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Resultsnoise removal

• Pepper and salt

White gaussian
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ResultsImage Segmentation Enhancement
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Questions?

• Thanks