1. 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
Moravske Toplice, 2008
Alexander Shekhovtsov, Vaclav Hlavac, Prague

2. Motivation I 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
Alexander Shekhovtsov, Vaclav Hlavac, Prague

3.   ? ? Motivation II 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
Alexander Shekhovtsov, Vaclav Hlavac, Prague

4. Motivation II 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
Alexander Shekhovtsov, Vaclav Hlavac, Prague

5. Decompositions 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
Alexander Shekhovtsov, Vaclav Hlavac, Prague

6. Lower Bounds 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
Alexander Shekhovtsov, Vaclav Hlavac, Prague

7. Per-node Bounds Fix a node Compute 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
Alexander Shekhovtsov, Vaclav Hlavac, Prague

8. Per-node Bounds: Experiments
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
Alexander Shekhovtsov, Vaclav Hlavac, Prague

9. Discussion
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
Alexander Shekhovtsov, Vaclav Hlavac, Prague