Hierarchical Image Segmentation Based on Semidefinite Programming

  • Jens Keuchel
  • Matthias Heiler
  • Christoph Schnörr
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3175)


Image segmentation based on graph representations has been a very active field of research recently. One major reason is that pairwise similarities (encoded by a graph) are also applicable in general situations where prototypical image descriptors as partitioning cues are no longer adequate. In this context, we recently proposed a novel convex programming approach for segmentation in terms of optimal graph cuts which compares favorably with alternative methods in several aspects.

In this paper we present a fully elaborated version of this approach along several directions: first, an image preprocessing method is proposed to reduce the problem size by several orders of magnitude. Furthermore, we argue that the hierarchical partition tree is a natural data structure as opposed to enforcing multiway cuts directly. In this context, we address various aspects regarding the fully automatic computation of the final segmentation. Experimental results illustrate the encouraging performance of our approach for unsupervised image segmentation.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Zhu, S.-C.: Statistical modeling and conceptualization of visual patterns. IEEE Trans. Patt. Anal. Mach. Intell. 25(6), 691–712 (2003)CrossRefGoogle Scholar
  2. 2.
    van Cutsem, B. (ed.): Classification and Dissimilarity Analysis. Lecture Notes in Statistics, vol. 93. Springer, Heidelberg (1994)zbMATHGoogle Scholar
  3. 3.
    Hahn, U., Ramscar, M. (eds.): Similarity and Categorization. Oxford Univ. Press, Oxford (2001)Google Scholar
  4. 4.
    Hofmann, T., Buhmann, J.: Pairwise data clustering by deterministic annealing. IEEE Trans. Patt. Anal. Mach. Intell. 19(1), 1–14 (1997)CrossRefGoogle Scholar
  5. 5.
    Puzicha, J., Buhmann, J.M.: Multiscale annealing for unsupervised image segmentation. Comp. Vision and Image Underst. 76(3), 213–230 (1999)CrossRefGoogle Scholar
  6. 6.
    Mohar, B., Poljak, S.: Eigenvalues in combinatorial optimization. In: Combinatorial and Graph-Theoretical Problems in Linear Algebra. IMA Vol. Math. Appl., vol. 50, pp. 107–151. Springer, Heidelberg (1993)Google Scholar
  7. 7.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Patt. Anal. Mach. Intell. 22(8), 888–905 (2000)CrossRefGoogle Scholar
  8. 8.
    Keuchel, J., Schnörr, C., Schellewald, C., Cremers, D.: Binary partitioning, perceptual grouping, and restoration with semidefinite programming. IEEE Trans. Patt. Anal. Mach. Intell. 25(11), 1364–1379 (2003)CrossRefGoogle Scholar
  9. 9.
    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: Proc. 8th Int. Conf. Computer Vision (ICCV), vol. 2, pp. 416–423. IEEE Comp. Soc, Los Alamitos (2001)Google Scholar
  10. 10.
    Comaniciu, D., Meer, P.: Mean shift: A robust approach toward feature space analysis. IEEE Trans. Patt. Anal. Mach. Intell. 24(5), 603–619 (2002)CrossRefGoogle Scholar
  11. 11.
    Ren, X., Malik, J.: Learning a classification model for segmentation. In: Proc. 9th Int. Conf. Computer Vision (ICCV), pp. 10–17. IEEE Comp. Soc, Los Alamitos (2003)CrossRefGoogle Scholar
  12. 12.
    Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.): Handbook of Semidefinite Programming. International series in operations research & management science, vol. 27. Kluwer Acad. Publ., Boston (2000)Google Scholar
  13. 13.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42(6), 1115–1145 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Malik, J., Belongie, S., Leung, T., Shi, J.: Contour and texture analysis for image segmentation. Int. J. Comp. Vision 43(1), 7–27 (2001)zbMATHCrossRefGoogle Scholar
  15. 15.
    Alpert, C.J., Kahng, A.B., Yao, S.-Z.: Spectral partitioning with multiple eigenvectors. Discrete Applied Math. 90, 3–26 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Yu, S.X., Shi, J.: Multiclass spectral clustering. In: Proc. 9th Int. Conf. Computer Vision (ICCV), pp. 313–319. IEEE Comp. Soc., Los Alamitos (2003)CrossRefGoogle Scholar
  17. 17.
    Fowlkes, C., Belongie, S., Chung, F., Malik, J.: Spectral grouping using the Nyström method. IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 214–225 (2004)CrossRefGoogle Scholar
  18. 18.
    Keuchel, J., Schnörr, C.: Efficient graph cuts for unsupervised image segmentation using probabilistic sampling and SVD-based approximation. In: 3rd Internat. Workshop on Statist. and Comput. Theories of Vision, Nice, France (2003)Google Scholar
  19. 19.
    Martin, D., Fowlkes, C., Malik, J.: Learning to detect natural image boundaries using local brightness, color, and texture cues. IEEE Trans. Patt. Anal. Mach. Intell. 26(5), 530–549 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jens Keuchel
    • 1
  • Matthias Heiler
    • 1
  • Christoph Schnörr
    • 1
  1. 1.CVGPR-Group, Dept. Math. and Comp. ScienceUniversity of MannheimMannheimGermany

Personalised recommendations