Generalized Roof Duality for Multi-Label Optimization: Optimal Lower Bounds and Persistency
Conference paper
Abstract
We extend the concept of generalized roof duality from pseudo-boolean functions to real-valued functions over multi-label variables. In particular, we prove that an analogue of the persistency property holds for energies of any order with any number of linearly ordered labels. Moreover, we show how the optimal submodular relaxation can be constructed in the first-order case.
Keywords
multi-label higher-order roof duality MRF computer vision Download
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References
- 1.Kolmogorov, V., Zabih, R.: What energy functions can be minimized via graph cuts? IEEE Transactions on Pattern Analysis and Machine Intelligence 24, 657–673 (2004)Google Scholar
- 2.Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence 23, 1222–1239 (2001)CrossRefGoogle Scholar
- 3.Ishikawa, H.: Exact optimization for Markov random fields with convex priors. IEEE Transactions on Pattern Analysis and Machine Intelligence 25, 1333–1336 (2003)CrossRefGoogle Scholar
- 4.Schlesinger, D., Flach, B.: Transforming an arbitrary minsum problem into a binary one. Technical report, TU Dresden (2006)Google Scholar
- 5.Schlesinger, D.: Exact Solution of Permuted Submodular MinSum Problems. In: Yuille, A.L., Zhu, S.-C., Cremers, D., Wang, Y. (eds.) EMMCVPR 2007. LNCS, vol. 4679, pp. 28–38. Springer, Heidelberg (2007)CrossRefGoogle Scholar
- 6.Ishikawa, H.: Transformation of general binary MRF minimization to the first order case. IEEE Transactions on Pattern Analysis and Machine Intelligence 33, 1234–1249 (2011)CrossRefGoogle Scholar
- 7.Gallagher, A.C., Batra, D., Parikh, D.: Inference for order reduction in Markov random fields. In: IEEE Conference on Computer Vision and Pattern Recognition (2011)Google Scholar
- 8.Fix, A., Grubner, A., Boros, E., Zabih, R.: A graph cut algorithm for higher-order Markov random fields. In: IEEE International Conference on Computer Vision, Barcelona (2011)Google Scholar
- 9.Kahl, F., Strandmark, P.: Generalized roof duality for pseudo-boolean optimization. In: IEEE International Conference on Computer Vision (2011)Google Scholar
- 10.Kahl, F., Strandmark, P.: Generalized roof duality. Discrete Applied Mathematics (2012)Google Scholar
- 11.Hammer, P.L., Hansen, P., Simeone, B.: Roof duality, complementation and persistency in quadratic 0-1 optimization. Math. Programming 28, 121–155 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
- 12.Boros, E., Hammer, P.L., Sun, R., Tavares, G.: A max-flow approach to improved lower bounds for quadratic unconstrained binary optimization (QUBO). Discrete Optimization 5, 501–529 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
- 13.Boros, E., Hammer, P.L., Tavares, G.: Preprocessing of unconstrained quadratic binary optimization. RUTCOR Research Report RRR 10-2006, Rutgers University (2006)Google Scholar
- 14.Rother, C., Kolmogorov, V., Lempitsky, V., Szummer, M.: Optimizing binary MRFs via extended roof duality. In: IEEE Conference on Computer Vision and Pattern Recognition, Minneapolis, Minnesota (2007)Google Scholar
- 15.Kolmogorov, V.: Generalized roof duality and bisubmodular functions. Discrete Applied Mathematics 160, 416–426 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
- 16.Kohli, P., Shekhovtsov, A., Rother, C., Kolmogorov, V., Torr, P.: On partial optimality in multi-label MRFs. In: Proceedings of the 25th International Conference on Machine Learning, pp. 480–487 (2008)Google Scholar
- 17.Flach, B., Schlesinger, D.: Best labeling search for a class of higher order gibbs models. Pattern Recognition and Image Analysis 14, 249–254 (2004)Google Scholar
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