Transcript Слайд 1 - Center For Machine Perception (Cmp)

A Lower Bound by One-against-all Decomposition for Potts
Model Energy Minimization
Alexander Shekhovtsov and Václav Hlaváč
Czech Technical University in Prague
Faculty of Electrical Engineering, Department of Cybernetics
Center for Machine Perception
Czech Republic
[email protected], [email protected]
Moravske Toplice, 2008
Motivation I
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 Energy Minimization Problem
Denoision, Boykov01
Stereo, Boykov01
Segmentation, Kovtun03
 NP-hard; Many algorithms (Schlesinger76, Pearl88, Boykov01,
Wainwright03, Kolmogorov05, Komodakis05, Schlesinger07).
 Algorithms  LP-relaxation; Suboptimal LP solvers.
 A faster LP solver for Potts model?
Alexander Shekhovtsov & Vaclav Hlavac, Prague 2008
Motivation II
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 Potts Model
Minimize the number
of steps
NP-hard for 3 labels
 For 2 labels, it is solvable exactly by a min-cut / max-flow algorithm.
 A natural heuristic: solve only 2 label problems
?

Alexander Shekhovtsov & Vaclav Hlavac, Prague 2008

?
Motivation II
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 Heuristic of Fangfang Lu et al. ACCV 2007
Fix labels in the areas
where labeling is
consistent
Is not guaranteed to
be correct
 Can we propose a method which would fix provably optimal labels?
Alexander Shekhovtsov & Vaclav Hlavac, Prague 2008
Decompositions
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 Idea of decompositions by Wainwright03 (trees).
 We propose a new kind of decomposition (one-against-all):
=
+
+
is equivalent to a binary problem (2 labels).
Solvable exactly by a single graph cut.
Alexander Shekhovtsov & Vaclav Hlavac, Prague 2008
Lower Bounds
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 The decomposition is not unique:
=
Free variables:
+
+
,
 Theorem. Problem (*) is equivalent to standard LP-relaxation.
 Coordinate ascent algorithm for (*).
Alexander Shekhovtsov & Vaclav Hlavac, Prague 2008
Per-node Bounds
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 Fix a node
 Comput
e
 If
- not optimal.
 We obtain bounds of this type for free in our algorithm
 If only one label remains in a pixel then we say that it is a part of any
optimal solution.
Alexander Shekhovtsov & Vaclav Hlavac, Prague 2008
Per-node Bounds: Experiments
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 Sample random problems: 10 x 10 grid graph with 5 labels.
 Compute
, plot empirical estimate of
 Problem parameters are sampled uniformly – almost no evidence for
optimal choice.
 Real problems should be easier.
Alexander Shekhovtsov & Vaclav Hlavac, Prague 2008
Discussion
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 Our algorithm gets stuck in suboptimal points (satisfy necessary
conditions only but not sufficient ones).
 We don’t know if it is faster than other algorithms.
 Testing on real problems has to be performed.
 We tested a BnB solver based on our bounds. Small problems were
solved exactly. It is important to have the ground truth.
Thank You
Alexander Shekhovtsov & Vaclav Hlavac, Prague 2008