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Planar Cycle Covering Graphs for inference in MRFS

Planar Cycle Covering Graphs for inference in MRFS . The Typhon Algorithm A New Variational Approach to Ground S tate Computation in Binary Planar Markov Random Fields by Julian Yarkony , Charless Fowlkes Alexander Ihler. Foreground/Background Segmentation. Background. Foreground.

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Planar Cycle Covering Graphs for inference in MRFS

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  1. Planar Cycle Covering Graphs for inference in MRFS The Typhon Algorithm A New Variational Approach to Ground State Computation in Binary Planar Markov Random Fields by Julian Yarkony, CharlessFowlkesAlexander Ihler

  2. Foreground/Background Segmentation Background Foreground Use appearance, edges, and prior information to segment image into foreground and background regions. Edge Information can be very useful when good models for foreground and background are unavailable. 2

  3. Binary MRFs and segmentation Min_X E(,\theta) Use a real delta or a 1 Cost to take on foreground Cost to disagree with neighbors

  4. When can we find the exact minimum? • Sub-modular Problems ( > 0) • Solve by reduction to graph cut [Boykov 2002] • Planar Problems without unary potentials ( =0) • Solve using a reduction to minimum cost perfect matching. [Kastyln 1969, Fourtin 1969, Schauldolph 2007]

  5. Trick for eliminating unary potentials

  6. Trick for eliminating unary potentials Problem: transformed graph may no longer be planar

  7. Planarity lost • Recall: perfect matching solution requires • No unary potentials • Planar

  8. Idea: duplicate field node to maintain planarity relaxation min_Xi E(Xi,theta) >= min_Xi,XfElb(Xi,Xf,theta,theta_if)

  9. TYPHON: Optimizing the Lower Bound • Solve using projected sub-gradient • To solve alternate between gradient step in and optimizing X • This optimization is CONVEX so this procedure is guaranteed to find global optima

  10. Sub-Gradient Update Use (1/N) Add parentheses • Each xi neighbors several copies of the field node • Optimization drives the xf towards agreement • Preserve µif = µi Old value Step size Mean disagreement New value Disagreement Call this X_f3 etc.

  11. Sub-Gradient Update Use (1/N) Add parentheses • Each xi neighbors several copies of the field node • Optimization drives the xf towards agreement • Preserve µif = µi Old value Step size Mean disagreement New value Disagreement Call this X_f3 etc. 0 1 0 0 1

  12. Sub-Gradient Update Use (1/N) Add parentheses • Each duplicated edge is modified to encourage that all copies agree with x, or all copies disagree • Nodes that disagree have their cost increase • Nodes that agree have their cost decrease Old value Step size Mean disagreement New value Disagreement Call this X_f3 etc. 0 1 0 0 1

  13. Convergence of Upper and Lower Bounds during sub-gradient optimization Upper Bound Energy MAP Lower Bound Time

  14. Computing Upper Bound at Each Step Upper Bound I Ground State, Lower Bound Upper Bound II Upper bounds are obtained by using the configuration produced at any given time for all non-field nodes.

  15. Dual Decomposition • TRW decomposes MRF into a sum of trees [Wainwright 2005] • How many trees are needed? • Sufficient to choose a set of trees which cover each edge in the original graph at least once.

  16. Cycle Decomposition • - Cycles give a tighter bound than trees • - Collection of Cycles provides a tighter bound than trees. • How many cycles? • Lots!! • e.g. one way to ensure all cycles are covered is to include all triplets • - [Sontag 2008] uses cutting plane techniques to iteratively add cycles

  17. Lemma: Relaxation is tight for a single cycle = Reverse equality

  18. TYPHON relaxation covers all cycles • Every cycle of G is present somewhere in the new graph, with copies of the field node • That cycle and its field node copies are tight • TYPHON is at least as tight as the set of all cycle subproblems

  19. Experimental Results • Synthetic problem test set • “Easy”, “Medium”, and “Hard” parameters • Pairwise potentials drawn from uniform, U[-R,R] • Unary drawn from • Easy: 3.2*[-R,R] – strong local information • Medium: 0.8*[-R,R] • Hard: 0.2*[-R,R] – very weak local information • Compare to state of the art algorithms: • MPLP, [Sontag 2008] • RPM, [Schraudolph 2010] (R = 500)

  20. Duality Gap as a function of time • Size: 36x36 grids Easy Medium Hard

  21. Time Until Convergence Easy Medium Hard Runs which did not converge to the required tolerance are left off

  22. Conclusions • New variational bound for binary planar MRF’s • Equal to cycle decomposition. • Currently Applying to segmentation and extending to non-planar MRF’s and non-binary MRF’s

  23. Thank You

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