Exact Optimization of Discrete Constrained Total Variation Minimization Problems

  • Jérôme Darbon
  • Marc Sigelle
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3322)


This paper deals with the total variation minimization problem when the fidelity is either the L 2-norm or the L 1-norm. We propose an algorithm which computes the exact solution of these two problems after discretization. Our method relies on the decomposition of an image into its level sets. It maps the original problems into independent binary Markov Random Field optimization problems associated with each level set. Exact solutions of these binary problems are found thanks to minimum-cut techniques. We prove that these binary solutions are increasing and thus allow to reconstruct the solution of the original problems.


Monotone Property Markov Random Field Level Line Gradient Descent Algorithm Binary Problem 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)zbMATHCrossRefGoogle Scholar
  2. 2.
    Sauer, K., Bouman, C.: Bayesian estimation of transmission tomograms using segmentation based optimization. IEEE Nuclear Science 39, 1144–1152 (1992)CrossRefGoogle Scholar
  3. 3.
    Evans, L., Gariepy, R.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)zbMATHGoogle Scholar
  4. 4.
    Chambolle, A.: An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision 20, 89–97 (2004)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE PAMI 23, 1222–1239 (2001)Google Scholar
  6. 6.
    Pollak, I., Willsky, A., Huang, Y.: Nonlinear evolution equations as fast and exact solvers of estimation problems. To appear in IEEE Signal Processing (2004)Google Scholar
  7. 7.
    Amini, A., Weymouth, T., Jain, R.: Using dynamic programming for solving variational problems in vision. IEEE PAMI 12, 855–867 (1990)Google Scholar
  8. 8.
    Ishikawa, H.: Exact optimization for Markov random fields with priors. IEEE PAMI 25, 1333–1336 (2003)Google Scholar
  9. 9.
    Alliney, S.: A property of the minimum vectors of a regularizing functional defined by means of the absolute norm. IEEE Signal Processing 45, 913–917 (1997)CrossRefGoogle Scholar
  10. 10.
    Chan, T., Esedog̃lu, S.: Aspect of total variation regularized l 1 function approximation. Technical Report 7, UCLA (2004)Google Scholar
  11. 11.
    Nikolova, M.: Minimizers of cost-functions involving nonsmooth data-fidelity terms. SIAM J. Num. Anal. 40, 965–994 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Nguyen, H., Worring, M., van den Boomgaard, R.: Watersnakes: Energy-driven watershed segmentation. IEEE PAMI 23, 330–342 (2003)Google Scholar
  13. 13.
    Winkler, G.: Image Analysis, Random Fields and Dynamic Monte Carlo Methods. Applications of mathematics. Springer, Heidelberg (2003)Google Scholar
  14. 14.
    Guichard, F., Morel, J.: Mathematical morphology ”almost everywhere”. In: Proceedings of ISMM, pp. 293–303. Csiro Publishing (2002)Google Scholar
  15. 15.
    Propp, J.G., Wilson, D.B.: Exact sampling with coupled Markov chains and statistical mechanics. Random Structures and Algorithms 9, 223–252 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Geman, S., Geman, D.: Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE PAMI 6, 721–741 (1984)zbMATHGoogle Scholar
  17. 17.
    Greig, D., Porteous, B., Seheult, A.: Exact maximum a posteriori estimation for binary images. Journal of the Royal Statistics Society 51, 271–279 (1989)Google Scholar
  18. 18.
    Roy, S.: Stereo without epipolar lines: A maximum-flow formulation. International Journal of Computer Vision 34, 147–162 (1999)CrossRefGoogle Scholar
  19. 19.
    Kolmogorov, V., Zabih, R.: What energy can be minimized via graph cuts? IEEE PAMI 26, 147–159 (2004)Google Scholar
  20. 20.
    Boykov, Y., Kolmogorov, V.: An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. IEEE PAMI 26, 1124–1137 (2004)Google Scholar
  21. 21.
    Strong, D., Chan, T.: Edge preserving and scale-dependent properties of total variation regularization. Inverse Problem 19, 165–187 (2003)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. University Lecture Series, vol. 22 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jérôme Darbon
    • 1
    • 2
  • Marc Sigelle
    • 2
  1. 1.EPITA Research and Development Laboratory (LRDE)Le Kremlin-BicêtreFrance
  2. 2.ENST TSI / CNRS LTCI UMR 5141ParisFrance

Personalised recommendations