Level Lines as Global Minimizers of Energy Functionals in Image Segmentation

  • Charles Kervrann
  • Mark Hoebeke
  • Alain Trubuil
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1843)


We propose a variational framework for determining global minimizers of rough energy functionals used in image segmentation. Segmentation is achieved by minimizing an energy model, which is comprised of two parts: the first part is the interaction between the observed data and the model, the second is a regularity term. The optimal boundaries are the set of curves that globally minimize the energy functional. Our motivation comes from the observation that energy functionals are traditionally complex, for which it is usually difficult to precise global minimizers corresponding to “best” segmentations. Therefore, we focus on basic energy models, which global minimizers can be explicitly determined. In this paper, we prove that the set of curves that minimizes the image moment-based energy functionals is a family of level lines, i.e. the boundaries of level sets (connected components) of the image. For the completeness of the paper, we present a non-iterative algorithm for computing partitions with connected components. It leads to a sound initialization-free algorithm without any hidden parameter to be tuned.


Global Minimizer Image Segmentation Energy Model Segmentation Result Object Boundary 
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.


  1. 1.
    J. Beaulieu and M. Goldberg. Hierarchy in picture segmentation: a stepwise optimization approach. IEEE Trans. Patt. Anal. and Mach. Int., 11(2):150–163, 1989.CrossRefGoogle Scholar
  2. 2.
    A. Blake and A. Zisserman. Visual Reconstruction. MIT Press, Cambridge, Mass, 1987.Google Scholar
  3. 3.
    V. Caselles, B. Coll, and J. Morel. Topographic maps. preprint CEREMADE, 1997.Google Scholar
  4. 4.
    V. Caselles, R. Kimmel, and G. Sapiro. Geodesic active contours. Int J. Computer Vision, 22(1):61–79, 1997.zbMATHCrossRefGoogle Scholar
  5. 5.
    A. Chakraborty and J. Duncan. Game-theoretic integration for image segmentation. IEEE Trans. Patt. Anal. and Mach. Int., 21(1):12–30, 1999.CrossRefGoogle Scholar
  6. 6.
    L. Cohen. On active contour models and balloons. CVGIP: Image Understanding, 53(2):211–218, 1991.zbMATHCrossRefGoogle Scholar
  7. 7.
    L. Cohen. Deformable curves and surfaces in image analysis. In Int. Conf. Curves and Surfaces, Chamonix, France, 1996.Google Scholar
  8. 8.
    T. Darrell and A. Pentland. Cooperative robust estimation using layers of support. IEEE Trans. Patt. Anal. and Mach. Int., 17(5):474–487, 1995.CrossRefGoogle Scholar
  9. 9.
    X. Descombes and F. Kruggel. A markov pixon information approach for low-level image description. IEEE Trans. Patt. Anal. and Mach. Int., 21(6):482–494, 1999.CrossRefGoogle Scholar
  10. 10.
    S. Geman and D. Geman. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans. Patt. Anal. and Mach. Int., 6(6):721–741, 1984.zbMATHCrossRefGoogle Scholar
  11. 11.
    J. Istas. Statistics of processes and signal-image segmentation. University of Paris VII, 1997.Google Scholar
  12. 12.
    I. Jermyn and H. Ishikawa. Globally optimal regions and boundaries. In Int. Conf. on Comp. Vis., pages 904–910, Kerkyra, Greece, September 1999.Google Scholar
  13. 13.
    M. Kass, A. Witkin, and D. Terzopoulos. Snakes: active contour models. Int J. Computer Vision, 12(1):321–331, 1987.Google Scholar
  14. 14.
    C. Kervrann, M. Hoebeke, and A. Trubuil. A level line selection approach for object boundary estimation. In Int. Conf. on Comp. Vis., pages 963–968, Kerkyra, Greece, September 1999.Google Scholar
  15. 15.
    Y. Leclerc. Constructing simple stable descriptions for image partitioning. Int J. Computer Vision, 3:73–102, 1989.CrossRefGoogle Scholar
  16. 16.
    G. Matheron. Random Sets and Integral Geometry. John Wiley, New York, 1975.zbMATHGoogle Scholar
  17. 17.
    J. Møller and R. Waagepertersen. Markov connected component fields. Adv. in Applied Probability, pages 1–35, 1998.Google Scholar
  18. 18.
    P. Monasse and F. Guichard. Scale-space from a level line tree. In Int. Conf. on Scale-Space Theories Comp. Vis., pages 175–186, Kerkyra, Greece, September 1999.Google Scholar
  19. 19.
    J. Morel and S. Solimini. Variational Methods in Image Segmentation. Birkhauser, 1994.Google Scholar
  20. 20.
    D. Mumford. The Bayesian rationale for energy functionals. Geometry-Driven Diffusion in Domputer Vision, pages 141–153, Bart Romeny ed., Kluwer Academic, 1994.Google Scholar
  21. 21.
    D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and variational problems. Communication on Pure and applied Mathematics, 42(5):577–685, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    S. Osher and J. Sethian. Fronts propagating with curvature dependent speed: algorithms based on the hamilton-jacobi formulation. J. Computational Physics, 79:12–49, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    F. O’Sullivan and M. Qian. A regularized contrast statistic for object boundary estimation-implementation and statistical evaluation. IEEE Trans. Patt. Anal, and Mach. Int., 16(6):561–570, 1994.CrossRefGoogle Scholar
  24. 24.
    N. Paragios and R. Deriche. Coupled geodesic active regions for image segmentation: a level set approach. In Euro. Conf. on Comp. Vis., Dublin, Ireland, 2000.Google Scholar
  25. 25.
    T. Pavlidis and Y. Liow. Integrating region growing and edge detection. IEEE Trans. Patt. Anal. and Mach. Int., 12:225–233, 1990.CrossRefGoogle Scholar
  26. 26.
    C. Schnörr. A study of a convex variational diffusion approach for image segmentation and feature extraction. J. Math. Imaging and Vision, 3(8):271–292, 1998.CrossRefGoogle Scholar
  27. 27.
    J. Wang. Stochastic relaxation on partitions with connected components and its application to image segmentation. IEEE Trans. Patt. Anal. and Mach. Int., 20(6):619–636, 1998.CrossRefGoogle Scholar
  28. 28.
    L. Younes. Calibrating parameters of cost functionals. In Euro. Conf. on Comp. Vis., Dublin, Ireland, 2000.Google Scholar
  29. 29.
    S. Zhu and A. Yuille. Region competition: unifying snakes, region growing, and bayes/MDL for multiband image segmentation. IEEE Trans. Patt. Anal. and Mach. Int., 18(9):884–900, 1996.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Charles Kervrann
    • 1
  • Mark Hoebeke
    • 1
  • Alain Trubuil
    • 1
  1. 1.INRA - Biométrie, Domaine de VilvertJouy-en-JosasFrance

Personalised recommendations