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A Trainable Graph Combination Scheme for Belief Propagation

A Trainable Graph Combination Scheme for Belief Propagation. Kai Ju Liu New York University. Images. Pairwise Markov Random Field. 4. 1. 2. 3. 5. Basic structure: vertices, edges. and observed value y i. Compatibility between states and observed values,.

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A Trainable Graph Combination Scheme for Belief Propagation

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  1. A Trainable Graph Combination Scheme for Belief Propagation Kai Ju Liu New York University

  2. Images

  3. Pairwise Markov Random Field 4 1 2 3 5 • Basic structure: vertices, edges

  4. and observed value yi • Compatibility between states and observed values, • Compatibility between neighboring vertices i and j, Pairwise Markov Random Field • Basic structure: vertices, edges • Vertex i has set of possible states Xi

  5. Marginal probability: Pairwise MRF: Probabilities • Joint probability: • Advantage: allows average over ambiguous states • Disadvantage: complexity exponential in number of vertices

  6. Belief Propagation 4 1 2 3 5

  7. Messages propagate information: Belief Propagation • Beliefs replace probabilities:

  8. When can we calculate beliefs exactly? When do beliefs equal probabilities? When is belief propagation efficient? Answer: Singly-Connected Graphs (SCG’s) • Graphs without loops • Messages terminate at leaf vertices • Beliefs equal probabilities • Complexity in previous example reduced from 13S5 to 24S2 BP: Questions

  9. Messages do not terminate 1 2 4 3 BP on Loopy Graphs • Energy approximation schemes [Freeman et al.] • Standard belief propagation • Generalized belief propagation • Standard belief propagation • Approximates Gibbs free energy of system by Bethe free energy • Iterates, requiring convergence criteria

  10. BP on Loopy Graphs • Tree-based reparameterization [Wainwright] • Reparameterizes distributions on singly-connected graphs • Convergence improved compared to standard belief propagation • Permits calculation of bounds on approximation errors

  11. BP-TwoGraphs • Eliminates iteration • Utilizes advantages of SCG’s

  12. Calculate beliefs on each set of SCG’s: • Select set of beliefs with minimum entropy BP-TwoGraphs • Consider loopy graph with n vertices • Select two sets of SCG’s that approximate the graph

  13. Rectangular grid of pixel vertices Hi: horizontal graphs Gi: vertical graphs BP-TwoGraphs on Images original graph vertical graph horizontal graph

  14. Image Segmentation add noise segment

  15. Image Segmentation Results

  16. Image Segmentation Revisited add noise ground truth max-flow ground truth

  17. Image Segmentation:Horizontal Graph Analysis

  18. Image Segmentation:Vertical Graph Analysis

  19. Rectangular grid of pixel vertices Hi: horizontal lines Gi: vertical lines BP-TwoLines original graph vertical line horizontal line

  20. Image Segmentation Results II

  21. Image Segmentation Results III

  22. Natural Image Segmentation

  23. Boundary-Based Image Segmentation: Window Vertices • Square 2-by-2 window of pixels • Each pixel has two states • foreground • background

  24. Boundary-Based Image Segmentation: Overlap

  25. Boundary-Based Image Segmentation: Graph

  26. Real Image Segmentation: Training

  27. Real Image Segmentation: Results

  28. Real Image Segmentation: Gorilla Results

  29. Conclusion • BP-TwoGraphs • Accurate and efficient • Extensive use of beliefs • Trainable parameters • Future work • Multiple states • Stereo • Image fusion

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