Skip to main content
SpringerLink
Log in

Automated Regularization Parameter Selection in Multi-Scale Total Variation Models for Image Restoration

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

Multi-scale total variation models for image restoration are introduced. The models utilize a spatially dependent regularization parameter in order to enhance image regions containing details while still sufficiently smoothing homogeneous features. The fully automated adjustment strategy of the regularization parameter is based on local variance estimators. For robustness reasons, the decision on the acceptance or rejection of a local parameter value relies on a confidence interval technique based on the expected maximal local variance estimate. In order to improve the performance of the initial algorithm a generalized hierarchical decomposition of the restored image is used. The corresponding subproblems are solved by a superlinearly convergent algorithm based on Fenchel-duality and inexact semismooth Newton techniques. The paper ends by a report on numerical tests, a qualitative study of the proposed adjustment scheme and a comparison with popular total variation based restoration methods.

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. USC-SIPI image database. University of Southern California. http://sipi.usc.edu/services/database/Database.html

  2. Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl. 10, 1217–1229 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Almansa, A., Ballester, C., Caselles, V., Haro, G.: A TV based restoration model with local constraints. J. Sci. Comput. 34(3), 209–236 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Andrews, H.C., Hunt, B.R.: Digital Image Restoration. Prentice Hall, New York (1977)

    Google Scholar 

  5. Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces. SIAM, Philadelphia (2005)

    Google Scholar 

  6. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations. Springer, New York (2002)

    MATH  Google Scholar 

  7. Bertalmio, M., Caselles, V., Rougé, B., Solé, A.: TV based image restoration with local constraints. J. Sci. Comput. 19, 95–122 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bovik, A.: Handbook of Image and Video Processing. Academic Press, San Diego (2000)

    MATH  Google Scholar 

  9. Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chambolle, A.: An algorithm for total variation minimization and application. J. Math. Imaging Vis. 20, 89–97 (2004)

    Article  MathSciNet  Google Scholar 

  11. Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chang, Q., Chern, I.-L.: Acceleration methods for total variation-based image denoising. SIAM J. Appl. Math. 25, 982–994 (2003)

    MathSciNet  MATH  Google Scholar 

  13. Dobson, D.C., Vogel, C.R.: Convergence of an iterative method for total variation denoising. SIAM J. Numer. Anal. 34, 1779–1791 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Classics Appl. Math., vol. 28. SIAM, Philadelphia (1999)

    MATH  Google Scholar 

  15. Facciolo, G., Almansa, A., Aujol, J.-F., Caselles, V.: Irregular to regular sampling, denoising and deconvolution. Multiscale Model. Simul. 7(4), 1574–1608 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Galatsanos, N.P., Ketsaggelos, A.K.: Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation. IEEE Trans. Image Process. 1, 322–336 (1992)

    Article  Google Scholar 

  17. Gilboa, G., Sochen, N., Zeevi, Y.Y.: Texture preserving variational denoising using an adaptive fidelity term. In: Proceeding of the IEEE Workshop on Variational, Geometric and Level Set Methods in Computer Vision, Nice, France, pp. 137–144 (2003)

    Google Scholar 

  18. Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston (1984)

    MATH  Google Scholar 

  19. Gumbel, E.J.: Les valeurs extrêmes des distributions statistiques. Ann. Inst. Henri Poincaré 5(2), 115–158 (1935)

    MathSciNet  MATH  Google Scholar 

  20. Hintermüller, M., Kunisch, K.: Total bounded variation regularization as bilaterally constrained optimization problem. SIAM J. Appl. Math. 64, 1311–1333 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hintermüller, M., Stadler, G.: An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration. SIAM J. Sci. Comput. 28(1), 1–23 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mood, A.: Introduction to the Theory of Statistics. McGraw-Hill, New York (1974)

    MATH  Google Scholar 

  23. Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4, 460–489 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Papoulis, A.: Probability, Random Variables, Stochastic Processes. McGraw-Hill, New York (1991)

    Google Scholar 

  25. Rudin, L.: MTV-multiscale total variation principle for a PDE-based solution to nonsmooth ill-posed problem. Technical report, Cognitech, Inc. Talk presented at the Workshop on Mathematical Methods in Computer Vision, University of Minnesota, 1995

  26. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    Article  MATH  Google Scholar 

  27. Rudin, W.: Functional Analysis. TATA McGraw-Hill Publishing Company LTD., Noida (1974)

    Google Scholar 

  28. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  29. Strong, D., Aujol, J.-F., Chan, T.: Scale recognition, regularization parameter selection, and Meyer’s G norm in total variation regularization. Technical report, UCLA, 2005

  30. Strong, D., Chan, T.: Spatially and scale adaptive total variation based regularization and anisotropic diffusion in image processing. Technical report, UCLA, 1996

  31. Strong, D., Chan, T.: Edge-preserving and scale-dependent properties of total variation regularization. Inverse Probl. 19, 165–187 (2003)

    Article  MathSciNet  Google Scholar 

  32. Tadmor, E., Nezzar, S., Vese, L.: A multiscale image representation using hierarchical (BV,L 2) decompositions. Multiscale Model. Simul. 2, 554–579 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tadmor, E., Nezzar, S., Vese, L.: Multiscale hierarchical decomposition of images with applications to deblurring, denoising and segmentation. Commun. Math. Sci. 6, 1–26 (2008)

    MathSciNet  Google Scholar 

  34. Vogel, C.R.: Computational Methods for Inverse Problems. Frontiers Appl. Math., vol. 23. SIAM, Philadelphia (2002)

    Book  MATH  Google Scholar 

  35. Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: From error visibility to structural similarity. IEEE Trans. Image Process. 13, 600–612 (2004)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Biomathematics and Biometry, Helmholtz Zentrum München, Ingolstädter Landstraße 1, 85764, Neuherberg, Germany

    Yiqiu Dong

  2. Department of Mathematics, Humboldt-University of Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Michael Hintermüller

  3. START-Project “Interfaces and Free Boundaries” and SFB “Mathematical Optimization and Applications in Biomedical Science”, Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010, Graz, Austria

    Michael Hintermüller & M. Monserrat Rincon-Camacho

Authors
  1. Yiqiu Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Hintermüller
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Monserrat Rincon-Camacho
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yiqiu Dong.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dong, Y., Hintermüller, M. & Rincon-Camacho, M.M. Automated Regularization Parameter Selection in Multi-Scale Total Variation Models for Image Restoration. J Math Imaging Vis 40, 82–104 (2011). https://doi.org/10.1007/s10851-010-0248-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-010-0248-9

Keywords

Search

Navigation