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A Lower Bound by One-against-all Decomposition for Potts Model Energy Minimization

A Lower Bound by One-against-all Decomposition for Potts Model Energy Minimization. Alexander Shekhovtsov and V á clav Hlav áč Czech Technical University in Prague Faculty of Electrical Engineering, Department of Cybernetics Center for Machine Perception Czech Republic

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A Lower Bound by One-against-all Decomposition for Potts Model Energy Minimization

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  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 shekhovt@cmp.felk.cvut.cz, hlavac@fel.cvut.cz Moravske Toplice, 2008

  2. Denoision, Boykov01 Stereo, Boykov01 Segmentation, Kovtun03 Motivation I • Energy Minimization Problem • 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

  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

  4. Motivation II • Heuristic of Fangfang Lu et al.ACCV2007 Fix labels in the areaswhere 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

  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

  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

  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

  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

  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

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