Journal of Mathematical Imaging and Vision

, Volume 26, Issue 3, pp 277–291

Image Restoration with Discrete Constrained Total Variation Part II: Levelable Functions, Convex Priors and Non-Convex Cases

Article

Abstract

In Part II of this paper we extend the results obtained in Part I for total variation minimization in image restoration towards the following directions: first we investigate the decomposability property of energies on levels, which leads us to introduce the concept of levelable regularization functions (which TV is the paradigm of). We show that convex levelable posterior energies can be minimized exactly using the level-independant cut optimization scheme seen in Part I. Next we extend this graph cut scheme to the case of non-convex levelable energies.We present convincing restoration results for images corrupted with impulsive noise. We also provide a minimum-cost based algorithm which computes a global minimizer for Markov Random Field with convex priors. Last we show that non-levelable models with convex local conditional posterior energies such as the class of generalized Gaussian models can be exactly minimized with a generalized coupled Simulated Annealing.

Keywords

total variation level sets convexity Markov Random fields graph cuts levelable functions 

Abbreviations

KAP

Kluwer Academic Publishers

compuscript

Electronically submitted article

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Ahuja, T. Magnanti, and J. Orlin, Network Flows: Theory, Algorithms and Applications. Prentice Hall, 1993.Google Scholar
  2. 2.
    S. Alliney, “An algorithm for the minimization of mixed l 1 and l 2 norms with application to bayesian estimation,” IEEE Transactions on Signal Processing, Vol. 42, No. 3, pp. 618–627, 1994.CrossRefGoogle Scholar
  3. 3.
    J. Besag, “On the statistical analysis of dirty pictures,” Journal of the Royal Statistics Society, Vol. 48, pp. 259–302, 1986.MathSciNetMATHGoogle Scholar
  4. 4.
    A. Blake and A. Zisserman, Visual Reconstruction. MIT Press, 1987.Google Scholar
  5. 5.
    C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Transactions on Transactions on Signal Processing, Vol. 2, No. 3, pp. 296–310, 1993.Google Scholar
  6. 6.
    S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.Google Scholar
  7. 7.
    Y. Boykov and V. Kolmogorov, “An experimental comparison of Min-Cut/Max-Flow algorithms for energy minimization in vision,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 26, No. 9, 1124–1137, 2004.CrossRefGoogle Scholar
  8. 8.
    Y. Boykov, O. Veksler, and R. Zabih, “Fast approximate energy minimization via graph cuts,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 23, No. 11, pp. 1222–1239, 2001.CrossRefGoogle Scholar
  9. 9.
    D. Coupier, A. Desolneux, and B. Ycart, “Image denoising by statistical area thresholding,” Journal of Mathematical Imaging and Vision, Vol. 22, No. 2–3, pp. 183–197, 2005.CrossRefMathSciNetGoogle Scholar
  10. 10.
    S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the bayesian restoration of images,” IEEE Pattern Analysis and Machine Intelligence, Vol. 6, No. 6, pp. 721–741, 1984.CrossRefMATHGoogle Scholar
  11. 11.
    G. Giorgi, A. Guerraggio, and J. Thierfelder, Mathematics of Optimization: Smooth and Nonsmooth Case. Elsevier Science, 2004.Google Scholar
  12. 12.
    D. Greig, B. Porteous, and A. Seheult, “Exact maximum a posteriori estimation for binary images,” Journal of the Royal Statistics Society, Vol. 51, No. 2, pp. 271–279, 1989.Google Scholar
  13. 13.
    F. Guichard and J. Morel, “Mathematical morphology “almost everywhere,”” in Proceedings of ISMM, 2002, pp. 293–303.Google Scholar
  14. 14.
    H. Ishikawa, “Exact optimization for Markov random fields with convex priors,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 25, No. 10, pp. 1333–1336, 2003.CrossRefGoogle Scholar
  15. 15.
    S. Iwata, L. Fleischer, and S. Fujishige, “A combinatorial strongly polynomial algorithm for minimizing submodular functions,” Journal of the ACM, Vol. 48, No. 4, pp. 761–777, 2001.CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    V. Kolmogorov and R. Zabih, “What energy can be minimized via graph cuts?” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 26, No. 2, pp. 147–159, 2004.CrossRefGoogle Scholar
  17. 17.
    M. Nikolova, “Local strong homogeneity of a regularized extimator,” SIAM Journal on Applied Mathematics, Vol. 61, pp. 633–658, 2000.CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    M. Nikolova, “A variational approach to remove outliers and impulse noise,” Journal of Mathematical Imaging and Vision, Vol. 20, pp. 99–120, 2004.CrossRefMathSciNetGoogle Scholar
  19. 19.
    S. Osher, A. Solé, and L. Vese, “Image decomposition and restoration using total variation minimization and the H −1 norm,” J. Mult. Model. and Simul., Vol. 1, No. 3, 2003.Google Scholar
  20. 20.
    I. Pollak, A. Willsky, and Y. Huang, “Nonlinear evolution equations as fast and exact solvers of estimation problems,” IEEE Transactions on Signal Processing, Vol. 53, No. 2, pp. 484–498, 2005.CrossRefMathSciNetGoogle Scholar
  21. 21.
    L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D., Vol. 60, pp. 259–268, 1992.CrossRefMATHGoogle Scholar
  22. 22.
    K. Sauer and C. Bouman, “Bayesian estimation of transmission tomograms using segmentation based optimization,” IEEE Transactions on Nuclear Science, Vol. 39, No. 4, pp. 1144–1152, 1992.CrossRefGoogle Scholar
  23. 23.
    B. Zalesky, “Efficient determination of Gibbs estimators with submodular energy functions,” http://www.citebase.org/cgi-bin/citations?id=oai:arXiv.org:math/0304041, 2003.

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.EPITA Research and Development Laboratory (LRDE)Le Kremlin-BicêtreFrance
  2. 2.École Nationale Supérieure des Télécommunications (ENST)ParisFrance
  3. 3.École Nationale Supérieure des Télécommunications (ENST) / LTCI CNRS UMR 5141ParisFrance

Personalised recommendations