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Issue No.10 - October (2010 vol.32)
pp: 1846-1857
Karteek Alahari , Oxford Brookes University, Oxford
Pushmeet Kohli , Microsoft Research Cambridge, Cambridge
Philip H.S. Torr , Oxford Brookes University, Oxford
ABSTRACT
In this paper, we present novel techniques that improve the computational and memory efficiency of algorithms for solving multilabel energy functions arising from discrete mrfs or crfs. These methods are motivated by the observations that the performance of minimization algorithms depends on: 1) the initialization used for the primal and dual variables and 2) the number of primal variables involved in the energy function. Our first method (dynamic \alpha-expansion) works by “recycling” results from previous problem instances. The second method simplifies the energy function by “reducing” the number of unknown variables present in the problem. Further, we show that it can also be used to generate a good initialization for the dynamic \alpha-expansion algorithm by “reusing” dual variables. We test the performance of our methods on energy functions encountered in the problems of stereo matching and color and object-based segmentation. Experimental results show that our methods achieve a substantial improvement in the performance of \alpha-expansion, as well as other popular algorithms such as sequential tree-reweighted message passing and max-product belief propagation. We also demonstrate the applicability of our schemes for certain higher order energy functions, such as the one described in [1], for interactive texture-based image and video segmentation. In most cases, we achieve a 10-15 times speed-up in the computation time. Our modified \alpha-expansion algorithm provides similar performance to Fast-PD [2], but is conceptually much simpler. Both \alpha-expansion and Fast-PD can be made orders of magnitude faster when used in conjunction with the “reduce” scheme proposed in this paper.
INDEX TERMS
Markov random fields, multilabel problems, energy minimization, approximate algorithms.
CITATION
Karteek Alahari, Pushmeet Kohli, Philip H.S. Torr, "Dynamic Hybrid Algorithms for MAP Inference in Discrete MRFs", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.32, no. 10, pp. 1846-1857, October 2010, doi:10.1109/TPAMI.2009.194
REFERENCES
 [1] P. Kohli, M.P. Kumar, and P.H.S. Torr, "${\rm P}^3$ & Beyond: Move Making Algorithms for Solving Higher Order Functions," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 31, no. 9, pp. 1645-1656, Sept. 2009. [2] N. Komodakis, G. Tziritas, and N. Paragios, "Fast, Approximately Optimal Solutions for Single and Dynamic MRFs," Proc. IEEE Int'l Conf. Computer Vision and Pattern Recognition, 2007. [3] Y. Boykov, O. Veksler, and R. Zabih, "Fast Approximate Energy Minimization via Graph Cuts," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 23, no. 11, pp. 1222-1239, Nov. 2001. [4] V. Kolmogorov and R. Zabih, "Multi-Camera Scene Reconstruction via Graph Cuts," Proc. European Conf. Computer Vision, vol. 3, pp. 82-96, 2002. [5] C. Rother, S. Kumar, V. Kolmogorov, and A. Blake, "Digital Tapestry," Proc. IEEE Int'l Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 589-596, 2005. [6] R. Szeliski, R. Zabih, D. Scharstein, O. Veksler, V. Kolmogorov, A. Agarwala, M.F. Tappen, and C. Rother, "A Comparative Study of Energy Minimization Methods for Markov Random Fields," Proc. European Conf. Computer Vision, vol. 2, pp. 16-29, 2006. [7] J.S. Yedidia, W.T. Freeman, and Y. Weiss, "Generalized Belief Propagation," Proc. Advances in Neural Information Processing Systems, pp. 689-695, 2000. [8] Y. Boykov and V. Kolmogorov, "An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 9, pp. 1124-1137, Sept. 2004. [9] D. Freedman and P. Drineas, "Energy Minimization via Graph Cuts: Settling What Is Possible," Proc. IEEE Int'l Conf. Computer Vision and Pattern Recognition, vol. 2, pp. 939-946, 2005. [10] H. Ishikawa, "Exact Optimization for Markov Random Fields with Convex Priors," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 25, no. 10, pp. 1333-1336, Oct. 2003. [11] V. Kolmogorov and R. Zabih, "What Energy Functions Can Be Minimized via Graph Cuts?" IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 2, pp. 147-159, Feb. 2004. [12] V. Kolmogorov, "Convergent Tree-Reweighted Message Passing for Energy Minimization," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 28, no. 10, pp. 1568-1583, Oct. 2006. [13] M.J. Wainwright, T. Jaakkola, and A.S. Willsky, "MAP Estimation via Agreement on Trees: Message-Passing and Linear Programming," IEEE Trans. Information Theory, vol. 51, no. 11, pp. 3697-3717, Nov. 2005. [14] P.F. Felzenszwalb and D.P. Huttenlocher, "Efficient Belief Propagation for Early Vision," Proc. IEEE Int'l Conf. Computer Vision and Pattern Recognition, pp. 261-268, 2004. [15] O. Juan and Y. Boykov, "Active Graph Cuts," Proc. IEEE Int'l Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 1023-1029, 2006. [16] P. Kohli and P.H.S. Torr, "Efficiently Solving Dynamic Markov Random Fields Using Graph Cuts," Proc. IEEE Int'l Conf. Computer Vision, vol. 2, pp. 922-929, 2005. [17] N. Komodakis and G. Tziritas, "A New Framework for Approximate Labeling via Graph Cuts," Proc. IEEE Int'l Conf. Computer Vision, pp. 1018-1025, 2005. [18] V. Kolmogorov and C. Rother, "Minimizing Non-Submodular Functions with Graph Cuts: A Review," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 29, no. 7, pp. 1274-1279, July 2007. [19] I. Kovtun, "Partial Optimal Labeling Search for A NP-Hard Subclass of (Max,+) Problems," Proc. DAGM Symp., pp. 402-409, 2003. [20] E. Boros and P.L. Hammer, "Pseudo Boolean Optimization," Discrete Applied Math., vol. 123, pp. 155-225, 2002. [21] Y. Boykov and M.-P. Jolly, "Interactive Graph Cuts for Optimal Boundary and Region Segmentation of Objects in ND Images," Proc. IEEE Int'l Conf. Computer Vision, vol. 1, pp. 105-112, 2001. [22] J. Shotton, J.M. Winn, C. Rother, and A. Criminisi, "TextonBoost: Joint Appearance, Shape and Context Modeling for Multi-Class Object Recognition and Segmentation," Proc. European Conf. Computer Vision, vol. 1, pp. 1-15, 2006. [23] R. Paget and I.D. Longstaff, "Texture Synthesis via a Noncausal Nonparametric Multiscale Markov Random Field," IEEE Trans. Image Processing, vol. 7, no. 6, pp. 925-931, June 1998. [24] S. Roth and M.J. Black, "Fields of Experts: A Framework for Learning Image Priors," Proc. IEEE Int'l Conf. Computer Vision and Pattern Recognition, vol. 2, pp. 860-867, 2005. [25] X. Lan, S. Roth, D.P. Huttenlocher, and M.J. Black, "Efficient Belief Propagation with Learned Higher-Order Markov Random Fields," Proc. European Conf. Computer Vision, vol. 2, pp. 269-282, 2006. [26] P. Kohli, L. Ladicky, and P.H.S. Torr, "Graph Cuts for Minimizing Robust Higher Order Potentials," Proc. IEEE Int'l Conf. Computer Vision and Pattern Recognition, 2008. [27] S. Iwata, S.T. McCormick, and M. Shigeno, "A Strongly Polynomial Cut Canceling Algorithm for the Submodular Flow Problem," Proc. Int'l Conf. Integer Programming and Combinatorial Optimization, pp. 259-272, 1999. [28] B.A. Zalesky, "Efficient Determination of Gibbs Estimators with Submodular Energy Functions," http://arxiv.org/abs/math0304041v1, 2003. [29] D. Schlesinger and B. Flach, "Transforming an Arbitrary Minsum Problem into a Binary One," Technical Report TUD-FI06-01, Dresden Univ. of Tech nology, Apr. 2006. [30] J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, 1998. [31] M.I. Schlesinger and V. Hlavac, Ten Lectures on Statistical and Structural Pattern Recognition. Kluwer Academic Publishers, 2002. [32] T. Werner, "A Linear Programming Approach to Max-Sum Problem: A Review," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 29, no. 7, pp. 1165-1179, July 2007. [33] N. Komodakis, N. Paragios, and G. Tziritas, "MRF Optimization via Dual Decomposition: Message-Passing Revisited," Proc. IEEE Int'l Conf. Computer Vision, 2007. [34] P. Kohli, A. Shekhovtsov, C. Rother, V. Kolmogorov, and P.H.S. Torr, "On Partial Optimality in Multi-Label Mrfs," Proc. Int'l Conf. Machine Learning, pp. 480-487, 2008. [35] I. Kovtun, "Image Segmentation Based on Sufficient Conditions for Optimality in NP-Complete Classes of Structural Labeling Problems," PhD dissertation, IRTC ITS Nat'l Academy of Science Ukraine, 2004. [36] D. Scharstein and R. Szeliski, "A Taxonomy and Evaluation of Dense Two-Frame Stereo Correspondence Algorithm," Int'l J. Computer Vision, vol. 47, pp. 7-42, 2002. [37] F. Schroff, A. Criminisi, and A. Zisserman, "Single-Histogram Class Models for Image Segmentation," Proc. Indian Conf. Computer Vision, Graphics and Image Processing, 2006. [38] S. Birchfield and C. Tomasi, "A Pixel Dissimilarity Measure that is Insensitive to Image Sampling," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 20, no. 4, pp. 401-406, Apr. 1998.