Potts Model, Parametric Maxflow and K-Submodular Functions

The problem of minimizing the Potts energy function frequently occurs in computer vision applications. One way to tackle this NP-hard problem was proposed by Kovtun [19, 20]. It identifies a part of an optimal solution by running k maxflow computations, where k is the number of labels. The number of "labeled" pixels can be significant in some applications, e.g. 50-93% in our tests for stereo. We show how to reduce the runtime to O(log k) maxflow computations (or one parametric maxflow computation). Furthermore, the output of our algorithm allows to speed-up the subsequent alpha expansion for the unlabeled part, or can be used as it is for time-critical applications. To derive our technique, we generalize the algorithm of Felzenszwalb et al. [7] for Tree Metrics. We also show a connection to k-sub modular functions from combinatorial optimization, and discuss k-sub modular relaxations for general energy functions.

[1]  Vladimir Kolmogorov,et al.  An experimental comparison of min-cut/max- flow algorithms for energy minimization in vision , 2001, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[2]  Václav Hlavác,et al.  On Partial Opimality by Auxiliary Submodular Problems , 2011, ArXiv.

[3]  B. Zalesky.  Network flow optimization for restoration of images , 2001 .

[4]  Pushmeet Kohli,et al.  Dynamic Hybrid Algorithms for MAP Inference in Discrete MRFs , 2010, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[5]  Pierre Hansen,et al.  Roof duality, complementation and persistency in quadratic 0–1 optimization , 1984, Math. Program..

[6]  R. Zabih,et al.  Efficient Graph-Based Energy Minimization Methods in Computer Vision , 1999 .

[7]  Nikos Komodakis,et al.  Approximate Labeling via Graph Cuts Based on Linear Programming , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[8]  VekslerOlga,et al.  Fast Approximate Energy Minimization via Graph Cuts , 2001 .

[9]  Ivan Kovtun,et al.  Partial Optimal Labeling Search for a NP-Hard Subclass of (max, +) Problems , 2003, DAGM-Symposium.

[10]  Anna Huber,et al.  Towards Minimizing k-Submodular Functions , 2012, ISCO.

[11]  Vladimir Kolmogorov,et al.  An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision , 2004, IEEE Trans. Pattern Anal. Mach. Intell..

[12]  Éva Tardos,et al.  Globally optimal pixel labeling algorithms for tree metrics , 2010, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[13]  Heiko Hirschmüller,et al.  Evaluation of Cost Functions for Stereo Matching , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[14]  Vladimir Kolmogorov,et al.  Generalized roof duality and bisubmodular functions , 2010, Discret. Appl. Math..

[15]  Dorit S. Hochbaum,et al.  An efficient algorithm for image segmentation, Markov random fields and related problems , 2001, JACM.

[16]  P. Kohli,et al.  Efficiently solving dynamic Markov random fields using graph cuts , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[17]  Richard Szeliski,et al.  High-accuracy stereo depth maps using structured light , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[18]  Stanislav Zivny,et al.  The Power of Linear Programming for Valued CSPs , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[19]  Antonin Chambolle,et al.  Total Variation Minimization and a Class of Binary MRF Models , 2005, EMMCVPR.

[20]  D. Scharstein,et al.  A Taxonomy and Evaluation of Dense Two-Frame Stereo Correspondence Algorithms , 2001, Proceedings IEEE Workshop on Stereo and Multi-Baseline Vision (SMBV 2001).

[21]  Olga Veksler,et al.  Fast Approximate Energy Minimization via Graph Cuts , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

[22]  Robert E. Tarjan,et al.  A Fast Parametric Maximum Flow Algorithm and Applications , 1989, SIAM J. Comput..

[23]  Andrew V. Goldberg,et al.  Experimental Evaluation of Parametric Max-Flow Algorithms , 2007, WEA.

[24]  Antoon Kolen,et al.  Tree network and planar rectilinear location theory , 1986 .

[25]  Nikos Komodakis,et al.  Performance vs computational efficiency for optimizing single and dynamic MRFs: Setting the state of the art with primal-dual strategies , 2008, Comput. Vis. Image Underst..

[26]  Jérôme Darbon,et al.  Image Restoration with Discrete Constrained Total Variation Part I: Fast and Exact Optimization , 2006, Journal of Mathematical Imaging and Vision.

[27]  Tomás Werner,et al.  A Linear Programming Approach to Max-Sum Problem: A Review , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.