Planar Cycle Covering Graphs for inference in MRFS
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Transcript of Planar Cycle Covering Graphs for inference in MRFS
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Planar Cycle Covering Graphs for inference in MRFS
The Typhon AlgorithmA New Variational Approach to Ground State Computation in Binary Planar
Markov Random Fieldsby
Julian Yarkony, Charless Fowlkes
Alexander Ihler
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Foreground/Background Segmentation
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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.
Foreground
Background
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Binary MRFs and segmentation
Cost to take on foreground
Cost to disagree with neighbors
1X 2X 3X
4X 5X 6X
7X 8X 9X
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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]
ij
i
Trick for eliminating unary potentials
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Trick for eliminating unary potentials
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Problem: transformed graph may no longer be planar
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Planarity lost
• Recall: perfect matching solution requires– No unary potentials– Planar
Idea: duplicate field node to maintain planarity
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relaxation
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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
if
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Sub-Gradient Update
Old valueNew value
Step sizeDisagreement
Mean disagreement
• Each xi neighbors several copies of the field node
• Optimization drives the xf towards agreement
• Preserve µif = µi
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Sub-Gradient Update
• Each xi neighbors several copies of the field node
• Optimization drives the xf towards agreement
• Preserve µif = µi
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Old valueNew value
Step sizeDisagreement
Mean disagreement
0
0 1
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Sub-Gradient Update
• 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
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Old valueNew value
Step sizeDisagreement
Mean disagreement
0
0 1
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Convergence of Upper and Lower Bounds during sub-gradient optimization
Ener
gy MAP
Time
Lower Bound
Upper Bound
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Computing Upper Bound at Each Step
Ground State, Lower Bound
Upper Bound II
Upper Bound I
Upper bounds are obtained by using the configuration produced at any given time for all non-field nodes.
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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.
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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
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Lemma: Relaxation is tight for a single cycle
=
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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
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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)
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Duality Gap as a function of time
• Size: 36x36 grids Easy Medium Hard
510 510
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Time Until Convergence
Easy Medium Hard
Runs which did not converge to the required tolerance are left off
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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
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Thank You