Advertisement

Image Labeling and Grouping by Minimizing Linear Functionals over Cones

  • Christian Schellewald
  • Jens Keuchel
  • Christoph Schnörr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2134)

Abstract

We consider energy minimization problems related to image labeling, partitioning, and grouping, which typically show up at mid-level stages of computer vision systems. A common feature of these problems is their intrinsic combinatorial complexity from an optimization point-of-view. Rather than trying to compute the global minimum - a goal we consider as elusive in these cases - we wish to design optimization approaches which exhibit two relevant properties: First, in each application a solution with guaranteed degree of suboptimality can be computed. Secondly, the computations are based on clearly defined algorithms which do not comprise any (hidden) tuning parameters.

In this paper, we focus on the second property and introduce a novel and general optimization technique to the field of computer vision which amounts to compute a suboptimal solution by just solving a convex optimization problem. As representative examples, we consider two binary quadratic energy functionals related to image labeling and perceptual grouping. Both problems can be considered as instances of a general quadratic functional in binary variables, which is embedded into a higher-dimensional space such that suboptimal solutions can be computed as minima of linear functionals over cones in that space (semidefi-nite programs). Extensive numerical results reveal that, on the average, suboptimal solutions can be computed which yield a gap below 5% with respect to the global optimum in case where this is known.

Keywords

Global Optimum Linear Functional Convex Optimization Problem Random Signal Perceptual Grouping 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Geman and D. Geman. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans. Patt. Anal. Mach. Intell., 6(6):721–741, 1984.zbMATHCrossRefGoogle Scholar
  2. 2.
    J.E. Besag. On the analysis of dirty pictures (with discussion). J. R. Statist. Soc. B, 48:259–302, 1986.zbMATHMathSciNetGoogle Scholar
  3. 3.
    P.B. Chou and C.M. Brown. Multimodal reconstruction and segmentation with markov random fields and hcf optimization. In Proc. DARPA Image Underst. Workshop, pages 214–221, Cambridge, Massachussetts, April 6–8 1988.Google Scholar
  4. 4.
    F. Heitz, P. Perez, and P. Bouthemy. Multiscale minimization of global energy functions in some visual recovery problems. Comp. Vis. Graph. Image Proc.: IU, 59(1):125–134, 1994.CrossRefGoogle Scholar
  5. 5.
    C.-h. Wu and P.C. Doerschuk. Cluster expansions for the deterministic computation of bayesian estimators based on markov random fields. IEEE Trans. Patt. Anal. Mach. Intell., 17(3):275–293, 1995.CrossRefGoogle Scholar
  6. 6.
    Y. Boykov, O. Veksler, and R. Zabih. Markov random fields with efficient approximations. In Proc. IEEE Conf. on Comp. Vision Patt. Recog. (CVPR’98), pages 648–655, Santa Barbara, California, 1998.Google Scholar
  7. 7.
    Y.G. Leclerc. Constructing simple stable descriptions for image partitioning. Int. J. of Comp. Vision, 3(1):73–102, 1989.CrossRefGoogle Scholar
  8. 8.
    A. Blake and A. Zisserman. Visual Reconstruction. MIT Press, 1987.Google Scholar
  9. 9.
    D. Geiger and F. Girosi. Parallel and deterministic algorithms from mrf’s: Surface reconstruction. IEEE Trans. Patt. Anal. Mach. Intell., 13(5):401–412, 1991.CrossRefGoogle Scholar
  10. 10.
    L. Herault and R. Horaud. Figure-ground discrimination: A combinatorial optimization approach. IEEE Trans. Patt. Anal. Mach. Intell., 15(9):899–914, 1993.CrossRefGoogle Scholar
  11. 11.
    S. Gold and A. Rangarajan. A graduated assignment algorithm for graph matching. IEEE Trans. Patt. Anal. Mach. Intell., 18(4):377–388, 1996.CrossRefGoogle Scholar
  12. 12.
    T. Hofmann and J. Buhmann. Pairwise data clustering by deterministic annealing. IEEE Trans. Patt. Anal. Mach. Intell., 19(1):1–14, 1997.CrossRefGoogle Scholar
  13. 13.
    M. Sato and S. Ishii. Bifurcations in mean-field-theory annealing. Physical Review E, 53(5):5153–5168, 1996.CrossRefGoogle Scholar
  14. 14.
    E. Aarts and J.K. Lenstra, editors. Local Search in Combinatorial Optimization, Chichester, 1997. Wiley & Sons.Google Scholar
  15. 15.
    C. Peterson and B. Söderberg. Artificial neural networks. In Aarts and Lenstra [14], chapter 7.Google Scholar
  16. 16.
    C. Schnörr. Unique reconstruction of piecewise smooth images by minimizing strictly convex non-quadratic functionals. J. of Math. Imag. Vision, 4:189–198, 1994.CrossRefGoogle Scholar
  17. 17.
    C. Schnörr. A study of a convex variational diffusion approach for image segmentation and feature extraction. J. of Math. Imag. and Vision, 8(3):271–292, 1998.zbMATHCrossRefGoogle Scholar
  18. 18.
    D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math., 42:577–685, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    L. Lovász and A. Schrijver. Cones of matrices and set-functions and 0-1 optimization. SI AM J. Optimization, 1(2): 166–190, 1991.zbMATHCrossRefGoogle Scholar
  20. 20.
    M.X. Goemans and D.P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42:1115–1145, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    M. Bertero, T. Poggio, and V. Torre. Ill-posed problems in early vision. Proc. IEEE, 76:869–889, 1988.CrossRefGoogle Scholar
  22. 22.
    G. Winkler. Image Analysis, Random Fields and Dynamic Monte Carlo Methods, volume 27 of Appl. of Mathematics. Springer-Verlag, Heidelberg, 1995.zbMATHGoogle Scholar
  23. 23.
    S. Sarkar and K.L. Boyer. Perceptual organization in computer vision: A review and a proposal for a classificatory structure. IEEE Tr. Systems, Man, and Cyb., 23(2):382–399, 1993.CrossRefGoogle Scholar
  24. 24.
    Y. Nesterov and A. Nemirovskii. Interior Point Polynomial Methods in Convex Programming. SIAM, 1994.Google Scholar
  25. 25.
    S.J. Benson, Y. Ye, and X. Zhang. Mixed linear and semidefinite programming for combinatorial and quadratic optimization. Optimiz. Methods and Software, 11&12:515–544, 1999.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Christian Schellewald
    • 1
  • Jens Keuchel
    • 1
  • Christoph Schnörr
    • 1
  1. 1.Computer Vision, Graphics, and Pattern Recognition Group Department of Mathematics and Computer ScienceUniversity of MannheimMannheimGermany

Personalised recommendations