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Generic Cuts: An Efficient Algorithm for Optimal Inference in Higher Order MRF-MAP

  • Chetan Arora
  • Subhashis Banerjee
  • Prem Kalra
  • S. N. Maheshwari
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7576)

Abstract

We propose a new algorithm called Generic Cuts for computing optimal solutions to 2 label MRF-MAP problems with higher order clique potentials satisfying submodularity. The algorithm runs in time O(2 k n 3) in the worst case (k is clique order and n is the number of pixels). A special gadget is introduced to model flows in a high order clique and a technique for building a flow graph is specified. Based on the primal dual structure of the optimization problem the notions of capacity of an edge and cut are generalized to define a flow problem. We show that in this flow graph max flow is equal to min cut which also is the optimal solution to the problem when potentials are submodular. This is in contrast to all prevalent techniques of optimizing Boolean energy functions involving higher order potentials including those based on reductions to quadratic potential functions which provide only approximate solutions even for submodular functions. We show experimentally that our implementation of the Generic Cuts algorithm is more than an order of magnitude faster than all algorithms including reduction based whose outputs on submodular potentials are near optimal.

Keywords

Higher Order MRF-MAP Submodular Function Minimization Optimal Algorithm 

References

  1. 1.
    Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell. 23, 1222–1239 (2001)CrossRefGoogle Scholar
  2. 2.
    Kleinberg, J., Tardos, E.: Approximation algorithms for classification problems with pairwise relationships: metric labeling and markov random fields. J. ACM 49, 616–639 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Komodakis, N., Tziritas, G.: Approximate labeling via graph cuts based on linear programming. IEEE Trans. Pattern Anal. Mach. Intell. 29, 1436–1453 (2007)CrossRefGoogle Scholar
  4. 4.
    Ishikawa, H.: Transformation of general binary MRF minimization to the first-order case. IEEE Trans. Pattern Anal. Mach. Intell. 33, 1234–1249 (2011)CrossRefGoogle Scholar
  5. 5.
    Kohli, P., Ladický, L., Torr, P.H.: Robust higher order potentials for enforcing label consistency. International Journal of Computer Vision 82, 302–324 (2009)CrossRefGoogle Scholar
  6. 6.
    Roth, S., Black, M.J.: Fields of experts. International Journal of Computer Vision 82, 205–229 (2009)CrossRefGoogle Scholar
  7. 7.
    Rother, C., Kohli, P., Feng, W., Jia, J.: Minimizing sparse higher order energy functions of discrete variables. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 1382–1389 (2009)Google Scholar
  8. 8.
    Woodford, O., Torr, P., Reid, I., Fitzgibbon, A.: Global stereo reconstruction under second order smoothness priors. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8 (2008)Google Scholar
  9. 9.
    Komodakis, N., Paragios, N.: Beyond pairwise energies: Efficient optimization for higher-order MRF’s. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 2985–2992 (2009)Google Scholar
  10. 10.
    Besag, J.: On the statistical analysis of dirty pictures. Journal of the Royal Statistical Society B-48, 259–302 (1986)MathSciNetGoogle Scholar
  11. 11.
    Koller, D., Friedman, N.: Probabilistic Graphical Models: Principles and Techniques. In: Adaptive Computation and Machine Learning. MIT Press (2009)Google Scholar
  12. 12.
    Kolmogorov, V., Rother, C.: Minimizing nonsubmodular functions with graph cuts-a review. IEEE Trans. Pattern Anal. Mach. Intell. 29, 1274–1279 (2007)CrossRefGoogle Scholar
  13. 13.
    Kahl, F., Strandmark, P.: Generalized roof duality for pseudo-boolean optimization. In: IEEE International Conference on Computer Vision, pp. 255–262 (2011)Google Scholar
  14. 14.
    Boros, E., Gruber, A.: On quadratization of pseudo-boolean functions. In: International Symposium on Artificial Intelligence and Mathematics (2012)Google Scholar
  15. 15.
    Fix, A., Gruber, A., Boros, E., Zabih, R.: A graph cut algorithm for higher-order markov random fields. In: IEEE International Conference on Computer Vision, pp. 1020–1027 (2011)Google Scholar
  16. 16.
    Freedman, D., Drineas, P.: Energy minimization via graph cuts: settling what is possible. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 939–946 (2005)Google Scholar
  17. 17.
    Kolmogorov, V., Zabin, R.: What energy functions can be minimized via graph cuts? IEEE Transactions on Pattern Analysis and Machine Intelligence 26, 147–159 (2004)CrossRefGoogle Scholar
  18. 18.
    Živný, S., Jeavons, P.G.: Classes of submodular constraints expressible by graph cuts. Constraints 15, 430–452 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Kolmogorov, V.: Generalized roof duality and bisubmodular functions. Discrete Applied Mathematics 160, 416–426 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Iwata, S., Orlin, J.B.: A simple combinatorial algorithm for submodular function minimization. In: Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1230–1237 (2009)Google Scholar
  21. 21.
    Kolmogorov, V.: Minimizing a sum of submodular functions. CoRR abs/1006.1990 (2010)Google Scholar
  22. 22.
    (Supplementary material)Google Scholar
  23. 23.
    Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19, 248–264 (1972)zbMATHCrossRefGoogle Scholar
  24. 24.
  25. 25.
  26. 26.
    Darwin framework, version 1.1.2, http://drwn.anu.edu.au/
  27. 27.
    Kolmogorov, V.: Convergent tree-reweighted message passing for energy minimization. IEEE Trans. Pattern Anal. Mach. Intell. 28, 1568–1583 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Chetan Arora
    • 1
  • Subhashis Banerjee
    • 1
  • Prem Kalra
    • 1
  • S. N. Maheshwari
    • 1
  1. 1.Indian Institute of Technology DelhiIndia

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