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

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Presentation transcript:

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

Images

Pairwise Markov Random Field Basic structure: vertices, edges

Pairwise Markov Random Field Basic structure: vertices, edges Vertex i has set of possible states X i and observed value y i Compatibility between states and observed values, Compatibility between neighboring vertices i and j,

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

Belief Propagation

Beliefs replace probabilities: Messages propagate information:

Belief Propagation Example

BP: Questions 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 13S 5 to 24S 2

BP on Loopy Graphs Messages do not terminate 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 12 43

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

BP-TwoGraphs Eliminates iteration Utilizes advantages of SCG’s

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

BP-TwoGraphs on Images Rectangular grid of pixel vertices H i : horizontal graphs G i : vertical graphs horizontal graph vertical graphoriginal graph

Image Segmentation add noise segment

Image Segmentation Results

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

Image Segmentation: Horizontal Graph Analysis

Image Segmentation: Vertical Graph Analysis

BP-TwoLines Rectangular grid of pixel vertices H i : horizontal lines G i : vertical lines horizontal line vertical lineoriginal graph

Image Segmentation Results II

Image Segmentation Results III

Natural Image Segmentation

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

Boundary-Based Image Segmentation: Overlap

Boundary-Based Image Segmentation: Graph

Real Image Segmentation: Training

Real Image Segmentation: Results

Real Image Segmentation: Gorilla Results

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