Toward Global Minimum through Combined Local Minima

  • Ho Yub Jung
  • Kyoung Mu Lee
  • Sang Uk Lee
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5305)


There are many local and greedy algorithms for energy minimization over Markov Random Field (MRF) such as iterated condition mode (ICM) and various gradient descent methods. Local minima solutions can be obtained with simple implementations and usually require smaller computational time than global algorithms. Also, methods such as ICM can be readily implemented in a various difficult problems that may involve larger than pairwise clique MRFs. However, their short comings are evident in comparison to newer methods such as graph cut and belief propagation. The local minimum depends largely on the initial state, which is the fundamental problem of its kind. In this paper, disadvantages of local minima techniques are addressed by proposing ways to combine multiple local solutions. First, multiple ICM solutions are obtained using different initial states. The solutions are combined with random partitioning based greedy algorithm called Combined Local Minima (CLM). There are numerous MRF problems that cannot be efficiently implemented with graph cut and belief propagation, and so by introducing ways to effectively combine local solutions, we present a method to dramatically improve many of the pre-existing local minima algorithms. The proposed approach is shown to be effective on pairwise stereo MRF compared with graph cut and sequential tree re-weighted belief propagation (TRW-S). Additionally, we tested our algorithm against belief propagation (BP) over randomly generated 30 ×30 MRF with 2 ×2 clique potentials, and we experimentally illustrate CLM’s advantage over message passing algorithms in computation complexity and performance.


  1. 1.
    Roth, S., Black, M.J.: Steerable random fields. In: ICCV (2007)Google Scholar
  2. 2.
    Roth, S., Black, M.J.: Field of experts: A framework for learning image priors. In: CVPR (2005)Google Scholar
  3. 3.
    Potetz, B.: Efficient belief propagation for vision using linear constraint nodes. In: CVPR (2007)Google Scholar
  4. 4.
    Kohli, P., Mudigonda, P., Torr, P.: p 3 and beyond: Solving energies with higher order cliques. In: CVPR (2007)Google Scholar
  5. 5.
    Rother, C., Kolmogorov, V., Minka, T., Blake, A.: Cosegmenation of image pairs by histogram matching- incorporating a global constraint into mrfs. In: CVPR (2006)Google Scholar
  6. 6.
    Lan, X., Roth, S., Huttenlocher, D., 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
  7. 7.
    Geman, S., Geman, D.: Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. PAMI 6 (1984)Google Scholar
  8. 8.
    Zhu, S.C., Liu, X.W., Wu, Y.N.: Exploring texture ensembles by efficent markov chain monte carlo: Toward a trichromacy theory of texture. PAMI 22(6) (2000)Google Scholar
  9. 9.
    Tu, Z., Zhu, S.C.: Image segmentation by data-driven markov chain monte carlo. PAMI 24 (2002)Google Scholar
  10. 10.
    Barbu, A., Zhu, S.C.: Generalizing swendsen-wang cut to sampling arbitrary posterior probabilities. PAMI 27 (2005)Google Scholar
  11. 11.
    Szeliski, R., Zabih, R., Scharstein, D., Veksler, O., Kolmogorov, V., Agarwala, A., Tappen, M., 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
  12. 12.
    Woodford, O.J., Reid, I.D., Torr, P.H.S., Fitzgibbon, A.W.: Field of experts for image-based rendering. BMVC (2006)Google Scholar
  13. 13.
    Jung, H.Y., Lee, K.M., Lee, S.U.: Window annealing over square lattice markov random field. ECCV (2008)Google Scholar
  14. 14.
    Besag, J.: On the statistical analysis of dirty pictures (with discussion). Journal of the Royal Statistical Society Series B 48 (1986)Google Scholar
  15. 15.
    Mignotte, M.: Nonparametric multiscale energy-based model and its application in some imagery problems. PAMI 26 (2004)Google Scholar
  16. 16.
  17. 17.
    Scharstein, D., Szeliski, R.: A taxonomy and evaluation of dense two-frame stereo correspondence algorithms. In: IJCV (2002)Google Scholar
  18. 18.
    Scharstein, D., Szeliski, R.: High-accuracy stereo depth maps using structured light. In: CVPR (2003)Google Scholar
  19. 19.
    Hirshmuller, H., Szeliski, R.: Evaluation of cost functions for stereo matching. In: CVPR (2007)Google Scholar
  20. 20.
    Scharstein, D., Pal, C.: Learning conditional random fields for stereo. In: CVPR (2007)Google Scholar
  21. 21.
    Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. PAMI 23 (2001)Google Scholar
  22. 22.
    Kolmogorov, V.: Convergent tree-reweighted message passing for energy minimization. PAMI 28 (2006)Google Scholar
  23. 23.
    Lempitsky, V., Rother, C., Blake, A.: Logcut - efficient graph cut optimization for markov random fields. In: ICCV (2007)Google Scholar
  24. 24.
    Crow, F.: Summed-area tables for texture mapping. SIGGRAPH (1984)Google Scholar
  25. 25.
    Tappen, M.F., Freeman, W.T.: Comparison of graph cuts with belief propagation for stereo, using identical mrf parameters. In: ICCV (2003)Google Scholar
  26. 26.
    Birchfield, S., Tomasi, C.: A pixel dissimilarity measure that is insensiitive to image samplin. PAMI 20 (1998)Google Scholar
  27. 27.
    Kolmogorov, V., Zabih, R.: What energy functions can be minimized via graph cuts? PAMI 26 (2004)Google Scholar
  28. 28.
    Boykov, Y., Kolmogorov, V.: An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. PAMI 26 (2004)Google Scholar
  29. 29.
    Wainwright, M.J., Jaakkola, T.S., Willsky, A.S.: Map estimation via agreement on trees: Message-passing and linear-programming approaches. IEEE Trans. Information Theory 51(11) (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ho Yub Jung
    • 1
  • Kyoung Mu Lee
    • 1
  • Sang Uk Lee
    • 1
  1. 1.Department of EECS, ASRISeoul National UniversitySeoulKorea

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