Publication 2011 Issue No. 6 - June Abstract - Transformation of General Binary MRF Minimization to the First-Order Case
Transformation of General Binary MRF Minimization to the First-Order Case
June 2011 (vol. 33 no. 6)
pp. 1234-1249
 ASCII Text x Hiroshi Ishikawa, "Transformation of General Binary MRF Minimization to the First-Order Case," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 33, no. 6, pp. 1234-1249, June, 2011.
 BibTex x @article{ 10.1109/TPAMI.2010.91,author = {Hiroshi Ishikawa},title = {Transformation of General Binary MRF Minimization to the First-Order Case},journal ={IEEE Transactions on Pattern Analysis and Machine Intelligence},volume = {33},number = {6},issn = {0162-8828},year = {2011},pages = {1234-1249},doi = {http://doi.ieeecomputersociety.org/10.1109/TPAMI.2010.91},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on Pattern Analysis and Machine IntelligenceTI - Transformation of General Binary MRF Minimization to the First-Order CaseIS - 6SN - 0162-8828SP1234EP1249EPD - 1234-1249A1 - Hiroshi Ishikawa, PY - 2011KW - Energy minimizationKW - pseudo-Boolean functionKW - higher-order MRFsKW - graph cuts.VL - 33JA - IEEE Transactions on Pattern Analysis and Machine IntelligenceER -
Hiroshi Ishikawa, Waseda University, Tokyo
We introduce a transformation of general higher-order Markov random field with binary labels into a first-order one that has the same minima as the original. Moreover, we formalize a framework for approximately minimizing higher-order multilabel MRF energies that combines the new reduction with the fusion-move and QPBO algorithms. While many computer vision problems today are formulated as energy minimization problems, they have mostly been limited to using first-order energies, which consist of unary and pairwise clique potentials, with a few exceptions that consider triples. This is because of the lack of efficient algorithms to optimize energies with higher-order interactions. Our algorithm challenges this restriction that limits the representational power of the models so that higher-order energies can be used to capture the rich statistics of natural scenes. We also show that some minimization methods can be considered special cases of the present framework, as well as comparing the new method experimentally with other such techniques.

[1] E. Boros, P.L. Hammer, R. Sun, and G. Tavares, “A Max-Flow Approach to Improved Lower Bounds for Quadratic Unconstrained Binary Optimization (QUBO),” Discrete Optimization, vol. 5, no. 2, pp. 501-529, 2008.
[2] E. Boros and P.L. Hammer, “Pseudo-Boolean Optimization,” Discrete Applied Math., vol. 123, nos. 1-3, pp. 155-225, 2002.
[3] E. Boros, P.L. Hammer, and G. Tavares, “Preprocessing of Unconstrained Quadratic Binary Optimization,” RUTCOR Research Report 10-2006, Rutgers Center for Operations Research, Rutgers Univ., Apr. 2006.
[4] 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.
[5] D. Cremers and L. Grady, “Statistical Priors for Efficient Combinatorial Optimization via Graph Cuts,” Proc. European Conf. Computer Vision, vol. 3, pp. 263-274, 2006.
[6] P. Felzenszwalb and D. Huttenlocher, “Efficient Belief Propagation for Early Vision,” Int'l. J. Computer Vision, vol. 70, pp. 41-54, 2006.
[7] D. Freedman and P. Drineas, “Energy Minimization via Graph Cuts: Settling What Is Possible,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, vol. 2, pp. 939-946, 2005.
[8] P.L. Hammer, P. Hansen, and B. Simeone, “Roof Duality, Complementation and Persistency in Quadratic 0-1 Optimization,” Math. Programming, vol. 28, pp. 121-155, 1984.
[9] P.L. Hammer and S. Rudeanu, Boolean Methods in Operations Research and Related Areas. Springer-Verlag, 1968.
[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] H. Ishikawa, “Higher-Order Clique Reduction in Binary Graph Cut,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, pp. 2993-3000, 2009.
[12] H. Ishikawa, “Higher-Order Gradient Descent by Fusion-Move Graph Cut,” Proc. IEEE Int'l Conf. Computer Vision, pp. 568-574, 2009.
[13] H. Ishikawa and D. Geiger, “Rethinking the Prior Model for Stereo,” Proc. European Conf. Computer Vision, vol. 3, pp. 526-537, 2006.
[14] P. Kohli, L. Ladicky, and P.H.S. Torr, “Robust Higher Order Potentials for Enforcing Label Consistency,” Int'l J. Compter Vision, vol. 82, no. 3, pp. 302-324, 2009.
[15] P. Kohli, M.P. Kumar, and P.H.S. Torr, “${\cal P}^3$ & 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.
[16] 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.
[17] 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.
[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] N. Komodakis and N. Paragios, “Beyond Pairwise Energies: Efficient Optimization for Higher-Order MRFs,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 2009.
[20] J.D. Lafferty, A. McCallum, and F. Pereira, “Conditional Random Fields: Probabilistic Models for Segmenting and Labeling Sequence Data,” Proc. Int'l Conf. Machine Learning, pp. 282-289, 2001.
[21] 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.
[22] V. Lempitsky, C. Rother, and A. Blake, “LogCut—Efficient Graph Cut Optimization for Markov Random Fields,” Proc. IEEE Int'l Conf. Computer Vision, 2007.
[23] V. Lempitsky, S. Roth, and C. Rother, “FusionFlow: Discrete-Continuous Optimization for Optical Flow Estimation,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 2008.
[24] D. Martin, C. Fowlkes, D. Tal, and J. Malik, “A Database of Human Segmented Natural Images and Its Application to Evaluating Segmentation Algorithms and Measuring Ecological Statistics,” Proc. IEEE Int'l Conf. Computer Vision, pp. 416-423, 2001.
[25] T. Meltzer, C. Yanover, and Y. Weiss, “Globally Optimal Solutions for Energy Minimization in Stereo Vision Using Reweighted Belief Propagation,” Proc. IEEE Int'l Conf. Computer Vision, pp. 428-435, 2005.
[26] G.L. Nemhauser, L.A. Wolsey, and M.L. Fisher, “An Analysis of Approximations for Maximizing Submodular Set Functions,” Math. Programming, vol. 14, no. 1, pp. 265-294, 1978.
[27] 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.
[28] B. Potetz, “Efficient Belief Propagation for Vision Using Linear Constraint Nodes,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 2007.
[29] I.G. Rosenberg, “Reduction of Bivalent Maximization to the Quadratic Case,” Cahiers du Centre d'Etudes de Recherche Operationnelle, vol. 17, pp. 71-74, 1975.
[30] S. Roth and M.J. Black, “Fields of Experts: A Framework for Learning Image Priors,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, vol. 2, pp. 860-867, 2005.
[31] C. Rother, V. Kolmogorov, V. Lempitsky, and M. Szummer, “Optimizing Binary MRFs via Extended Roof Duality,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 2007.
[32] C. Rother, V. Kolmogorov, T. Minka, and A. Blake, “Cosegmentation of Image Pairs by Histogram Matching—Incorporating a Global Constraint into MRFs,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, pp. 993-1000, 2006.
[33] C. Rother, P. Kohli, W. Feng, and J. Jia, “Minimizing Sparse Higher Order Energy Functions of Discrete Variables,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 2009.
[34] S. Ramalingam, P. Kohli, K. Alahari, and P.H.S. Torr, “Exact Inference in Multi-Label CRFs with Higher Order Cliques,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 2008.
[35] S. Vicente, V. Kolmogorov, and C. Rother, “Graph Cut Based Image Segmentation with Connectivity Priors,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 2008.
[36] O.J. Woodford, P.H.S. Torr, I.D. Reid, and A.W. Fitzgibbon, “Global Stereo Reconstruction under Second Order Smoothness Priors,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, 2008.

Index Terms:
Energy minimization, pseudo-Boolean function, higher-order MRFs, graph cuts.
Citation:
Hiroshi Ishikawa, "Transformation of General Binary MRF Minimization to the First-Order Case," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 33, no. 6, pp. 1234-1249, June 2011, doi:10.1109/TPAMI.2010.91