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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