Advertisement

Optimizing Binary MRFs with Higher Order Cliques

  • Asem M. Ali
  • Aly A. Farag
  • Georgy L. Gimel’farb
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5304)

Abstract

Widespread use of efficient and successful solutions of Computer Vision problems based on pairwise Markov Random Field (MRF) models raises a question: does any link exist between the pairwise and higher order MRFs such that the like solutions can be applied to the latter models? This work explores such a link for binary MRFs that allow us to represent Gibbs energy of signal interaction with a polynomial function. We show how a higher order polynomial can be efficiently transformed into a quadratic function. Then energy minimization tools for the pairwise MRF models can be easily applied to the higher order counterparts. Also, we propose a method to analytically estimate the potential parameter of the asymmetric Potts prior. The proposed framework demonstrates very promising experimental results of image segmentation and can be used to solve other Computer Vision problems.

Keywords

Markov Random Field Neighborhood System Clique Size Computer Vision Problem Quadratic Energy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Boykov, Y., Veksler, O., Zabih, R.: Fast Approximation Energy Minimization via Graph Cuts. IEEE Trans. PAMI 23(11), 1222–1239 (2001)CrossRefGoogle Scholar
  2. 2.
    Kolmogorov, V., Zabih, R.: What Energy Functions Can be Minimized via Graph Cuts? IEEE Trans. PAMI 26(2), 147–159 (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    Yedidia, J.S., Freeman, W.T., Weiss, Y.: Generalized Belief Propagation. In: NIPS, pp. 689–695 (2000)Google Scholar
  4. 4.
    Felzenszwalb, P.F., Huttenlocher, D.P.: Efficient Belief Propagation for Early Vision. Int. J. Computer Vision 70(1), 41–54 (2006)CrossRefGoogle Scholar
  5. 5.
    Kolmogorov, V.: Convergent Tree-Reweighted Message Passing for Energy Minimization. IEEE Trans. PAMI 28(10), 1568–1583 (2006)CrossRefGoogle Scholar
  6. 6.
    Wainwright, M.J., Jaakkola, T., Willsky, A.S.: Tree-Based Reparameterization for Approximate Inference on Loopy Graphs. In: NIPS, pp. 1001–1008 (2001)Google Scholar
  7. 7.
    Szeliski, R., Zabih, R., Scharstein, D., Veksler, O., Kolmogorov, V., Agarwala, A., Tappen, M.F., Rother, C.: A Comparative Study of Energy Minimization Methods for Markov Random Fields. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3952, pp. 16–29. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Lan, X., Roth, S., Huttenlocher, D.P., Black, M.J.: Efficient Belief Propagation with Learned Higher-Order Markov Random Fields. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3952, pp. 269–282. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Potetz, B.: Efficient Belief Propagation for Vision Using Linear Constraint Nodes. In: CVPR (2007)Google Scholar
  10. 10.
    Paget, R., Longstaff, I.D.: Texture Synthesis via a Noncausal Nonparametric Multiscale Markov Random Field. IEEE Trans. Image Processing 7(6), 925–931 (1998)CrossRefGoogle Scholar
  11. 11.
    Roth, S., Black, M.J.: Fields of Experts: A Framework for Learning Image Priors. In: CVPR, pp. 860–867 (2005)Google Scholar
  12. 12.
    Kohli, P., Kumar, M., Torr, P.: \(\mathcal{P}^3\) & beyond: Solving Energies with Higher Order Cliques. In: CVPR (2007)Google Scholar
  13. 13.
    Kolmogorov, V., Rother, C.: Minimizing Nonsubmodular Functions with Graph Cuts-A Review. IEEE Trans. PAMI 29(7), 1274–1279 (2007)CrossRefGoogle Scholar
  14. 14.
    Boros, E., Hammer, P.L.: Pseudo-Boolean Optimization. Discrete Appl. Math. 123(1-3), 155–225 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Rother, C., Kumar, S., Kolmogorov, V., Blake, A.: Digital Tapestry. In: CVPR, pp. 589–596 (2005)Google Scholar
  16. 16.
    Rother, C., Kolmogorov, V., Lempitsky, V.S., Szummer, M.: Optimizing Binary MRFs via Extended Roof Duality. In: CVPR (2007)Google Scholar
  17. 17.
    Geman, S., Geman, D.: Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images. IEEE Trans. PAMI 6, 721–741 (1984)CrossRefzbMATHGoogle Scholar
  18. 18.
    Rosenberg, I.G.: Reduction of Bivalent Maximization to The Quadratic Case. Cahiers du Centre d’Etudes de Recherche Operationnelle 17, 71–74 (1975)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Freedman, D., Drineas, P.: Energy Minimization via Graph Cuts: Settling What is Possible. In: CVPR, pp. 939–946 (2005)Google Scholar
  20. 20.
    Cunningham, W.: Minimum Cuts, Modular Functions, and Matroid Polyhedra. Networks 15, 205–215 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gimel’farb, G.L.: Image Textures and Gibbs Random Fields. Kluwer Academic Publishers, Dordrecht (1999)CrossRefzbMATHGoogle Scholar
  22. 22.
    Chen, C.C.: Markov Random Field Models in Image Analysis. PhD thesis, Michigan State University, East Lansing (1988)Google Scholar
  23. 23.
    Chen, S., Cao, L., J.L., Tang, X.: Iterative MAP and ML Estimations for Image Segmentation. In: CVPR (2007)Google Scholar
  24. 24.
    Carson, C., Belongie, S., Greenspan, H., Malik, J.: Blobworld: Image Segmentation Using Expectation-Maximization and Its Application to Image Querying. IEEE Trans. PAMI 24(8), 1026–1038 (2002)CrossRefGoogle Scholar
  25. 25.
    Martin, D., Fowlkes, C., Tal, D., Malik, J.: A Database of Human Segmented Natural Images and Its Application to Evaluating Segmentation Algorithms and Measuring Ecological Statistics. In: ICCV, pp. 416–423 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Asem M. Ali
    • 1
  • Aly A. Farag
    • 1
  • Georgy L. Gimel’farb
    • 2
  1. 1.Computer Vision and Image Processing LaboratoryUniversity of LouisvilleUSA
  2. 2.Department of Computer ScienceThe University of AucklandNew Zealand

Personalised recommendations