Skip to main content
SpringerLink
Log in

An automatic regularization parameter selection algorithm in the total variation model for image deblurring

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Image restoration is an inverse problem that has been widely studied in recent years. The total variation based model by Rudin-Osher-Fatemi (1992) is one of the most effective and well known due to its ability to preserve sharp features in restoration. This paper addresses an important and yet outstanding issue for this model in selection of an optimal regularization parameter, for the case of image deblurring. We propose to compute the optimal regularization parameter along with the restored image in the same variational setting, by considering a Karush Kuhn Tucker (KKT) system. Through establishing analytically the monotonicity result, we can compute this parameter by an iterative algorithm for the KKT system. Such an approach corresponds to solving an equation using discrepancy principle, rather than using discrepancy principle only as a stopping criterion. Numerical experiments show that the algorithm is efficient and effective for image deblurring problems and yet is competitive.

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Afonso, M., Boiucas-Dias, J., Figuereido, M.: An augmented lagrangian approach to the constrained optimization Formulation of Imaging Inverse Problems. IEEE Trans. Image Process. 20(3), 681–695 (2011)

    Article  MathSciNet  Google Scholar 

  3. Afonso, M., Boiucas-Dias, J., Figuereido, M.: Fast image recovery using variable splitting and constrained optimization. IEEE Trans. Image Process. 19(9), 2345–2356 (2010)

    Article  MathSciNet  Google Scholar 

  4. Bardsley, J., Goldes, J.: Regularization parameter selection and an efficient algorithm for total variation-regularized positron emission tomography. Num. Alg. 57, 255–271 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Blomgren, P., Chan, T., Color, T.V.: Total variation methods for restoration of vector-valued images. IEEE Trans. Image Process. 7(3), 304–309 (1998)

    Article  Google Scholar 

  6. Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brito, C., Chen, K.: Multigrid algorithm for high order denoising. SIAM J. Imaging Sci. 3(3), 363–389 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Chan, T., Osher, S.: The digital TV filter and nonlinear denoising. IEEE Trans. Image Process. 10(2), 231–241 (2001)

    Article  MATH  Google Scholar 

  12. Chan, T.F., Shen, J.: Image processing and analysis: variational, PDE, wavelet and stochastic methods. SIAM (2005)

  13. Chan, T., Tai, X.: Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients. J. Comp. Phy. 193, 40–66 (2003)

    Article  MathSciNet  Google Scholar 

  14. Chan, R., Chen, K.: A multilevel algorithm for simoultaneously denoising and deblurring images. Siam J. Sci. Comp. 32(2), 1043–1063 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chan, T., Chen, K.: An optimization-based multilevel algorithm for total variation image denoising. Multiscale Model. Simul. 5, 615–645 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dembo, R., Steihaug, T.: Truncated-Newton algorithms for large-scale unconstrained optimization. Math. Program. 26, 190–212 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  17. Dong, Y., Hintermüller, M., Rincon-Camacho, M.M.: Automated regularization parameter selection in multi-scale total variation models for image restoration. J. Math. Image Vis. 40(1), 83–104 (2011)

    Google Scholar 

  18. Hanke, M.: Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems. Num. Func. Anal. Optim. 18, 971–993 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hanke, M., Nagy, J.G., Plemmons, R.J.: Preconditioned iterative regularization for ill-posed problems. In: Reichel, L., Ruttan, A., Varga, R.S. (eds.) Numerical Linear Algebra, pp. 141–163. de Gruyter, Berlin (1993)

    Google Scholar 

  20. Hansen, P.C., Nagy, J.G., O’Leary, D.P.: Deblurring Images: Matrices, Spectra, and Filtering. SIAM publications, Philadelphia (2006)

    Book  Google Scholar 

  21. Hansen, P.C. et al.: Algorithms and software for total variation image reconstruction via first-order methods. Num. Alg. 53(1), 67–92 (2010)

    Article  MATH  Google Scholar 

  22. Huang, Y., Ng, M., Wen, Y.: A fast total variation minimization method for image restoration. Multiscale Model. Simul. 7, 774–795 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Martin, D.R., Reichel, L.: Projected Tikhonov regularization of large-scale discrete ill-posed problems, J. Sci. Comput. 56, 471–493 (2013)

    Google Scholar 

  24. Abad, J.O., Morigi, S., Reichel, L., Sgallari, F.: Alternating Krylov subspace image restoration methods. J. Comput. Appl. Math. 236, 2049–2062 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Loli Piccolomini, E., Zama, F.: A descent method for the regularization of ill-posed problems. Optim. Methods Softw. 20(4–5), 615–628 (2005)

    MATH  MathSciNet  Google Scholar 

  26. Loli Piccolomini, E., Zama, F.: An iterative algorithm for large size least-squares constrained regularization problems. Appl. Math. Comput. 217, 10343–10354 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  27. Reichel, L., Rodriguez, G.: Old and new parameter choice rules for discrete ill-posed problems. Num. Alg. 63, 65–87 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  30. Vogel, C.R., Oman, M.E.: Fast, robust total variation–based reconstruction of noisy, blurred images. IEEE Trans. Image Processing 7, 813–824 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  31. Vogel, C.R.: Computational Methods for Inverse Problems. SIAM Publications, USA (2002)

    Book  MATH  Google Scholar 

  32. Wen, Y., Chan, R.: Parameter selection for total variation based image restoration using discrepancy principle. IEEE TRans. Image Proc. 21(4), 1770–1781 (2012)

    Article  MathSciNet  Google Scholar 

  33. Wen, Y., Ng, M., Ching, W.K.: Iterative algorithms based on decoupling of deblurring and denoising for image restoration. Siam J. Sci. Comp. 30, 2655–2674 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  34. Zhang, J.P., Chen, K., Yu, B.: An iterative lagrange multiplier method for constrained total-variation-based image denoising. SIAM J. Numer. Anal. 50(3), 983–1003 (2012)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Mathematical Imaging Techniques (CMIT), and Department of Mathematical Sciences, University of Liverpool, Liverpool, UK

    K. Chen

  2. Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126, Bologna, Italy

    E. Loli Piccolomini & F. Zama

Authors
  1. K. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Loli Piccolomini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. Zama
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Loli Piccolomini.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, K., Piccolomini, E.L. & Zama, F. An automatic regularization parameter selection algorithm in the total variation model for image deblurring. Numer Algor 67, 73–92 (2014). https://doi.org/10.1007/s11075-013-9775-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-013-9775-y

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation