Skip to main content
SpringerLink
Log in

Non-smooth SOR for L 1-Fitting: Convergence Study and Discussion of Related Issues

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this article, the denoising of smooth (H 1-regular) images is considered. To reach this objective, we introduce a simple and highly efficient over-relaxation technique for solving the convex, non-smooth optimization problems resulting from the denoising formulation. We describe the algorithm, discuss its convergence and present the results of numerical experiments, which validate the methods under consideration with respect to both efficiency and denoising capability. Several issues concerning the convergence of an Uzawa algorithm for the solution of the same problem are also discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aujol, J.-F., Gilboa, G., Chan, T., Osher, S.: Structure-texture image decomposition—modeling, algorithms, and parameter selection. Int. J. Comput. Vis. 67, 111–136 (2006)

    Article  Google Scholar 

  2. Cea, J., Glowinski, R.: Sur des méthodes de minimisation par relaxation. In: Revue Française d’Automatique, Informatique, Recherche Opérationnelle, Série Rouge, pp. 5–32. (1973)

  3. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)

    MATH  Google Scholar 

  4. Glowinski, R.: Finite element methods for incompressible viscous flow. In: Numerical Methods for Fluids, Part 3. Handbook of Numerical Analysis, vol. IX, pp. 3–1176. North-Holland, Amsterdam (2003)

    Google Scholar 

  5. Glowinski, R., Lions, J., Trémolières, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)

    Book  MATH  Google Scholar 

  6. Gonzalez, R.C., Woods, R.E.: Digital Image Processing. Addison-Wesley, Reading (1993)

    Google Scholar 

  7. Grisvard, P.: Elliptic Problems in Non-Smooth Domains. Pitman, London (1985)

    Google Scholar 

  8. Huber, P.J.: Robust Statistics. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1981)

    MATH  Google Scholar 

  9. Jain, A.K.: Fundamentals of Digital Image Processing. Prentice-Hall, New York (1989)

    MATH  Google Scholar 

  10. Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)

    Article  MATH  Google Scholar 

  11. Kärkkäinen, T., Kunisch, K., Majava, K.: Denoising of smooth images using L 1-fitting. Computing 74, 353–376 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kärkkäinen, T., Kunisch, K., Tarvainen, P.: Augmented Lagrangian active set methods for obstacle problems. J. Optim. Theory Appl. 119, 499–533 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kärkkäinen, T., Toivanen, J.: Building blocks for odd-even multigrid with applications to reduced systems. J. Comput. Appl. Math. 131, 15–33 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Majava, K.: Optimization-based techniques for image restoration. PhD dissertation, University of Jyväskylä, Department of Mathematical Information Technology (2001)

  15. Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  16. Maunuksela, J., Myllys, M., Merikoski, J., Timonen, J., Kärkkäinen, T., Welling, M., Wijngaarden, R.: Determination of the stochastic evolution equation from noisy experimental data. Eur. Phys. J. B 33, 193–202 (2003)

    Article  Google Scholar 

  17. Nečas, J.: Les Méthodes Directes en Théorie des Équations Elliptiques. Masson, Paris (1967)

    Google Scholar 

  18. Nikolova, M.: Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers. SIAM J. Numer. Anal. 40, 965–994 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Rao, C.R.: Methodology based on the L 1-norm, in statistical inference. Sankhya Indian J. Stat. 50, 289–313 (1988)

    MATH  Google Scholar 

  20. Smith, S.W.: The Scientist and Engineer’s Guide to Digital Signal Processing. California Technical Publishing (1997)

  21. Valkonen, T.: Convergence of a SOR-Weiszfeld type algorithm for incomplete data sets. Numer. Funct. Anal. Optim. 27, 931–952 (2006)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Houston, 651 P.G. Hoffman Hall, Houston, TX, 77204-3008, USA

    R. Glowinski

  2. Department of Mathematical Information Technology, Universify of Jyväskylä, P.O. Box 35 (Agora), 40014, University of Jyväskylä, Finland

    T. Kärkkäinen, T. Valkonen & A. Ivannikov

Authors
  1. R. Glowinski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Kärkkäinen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. Valkonen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Ivannikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Kärkkäinen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Glowinski, R., Kärkkäinen, T., Valkonen, T. et al. Non-smooth SOR for L 1-Fitting: Convergence Study and Discussion of Related Issues. J Sci Comput 37, 103–138 (2008). https://doi.org/10.1007/s10915-008-9229-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-008-9229-1

Keywords

Search

Navigation