Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Noise-reducing cascadic multilevel methods for linear discrete ill-posed problems

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

Abstract

Cascadic multilevel methods for the solution of linear discrete ill-posed problems with noise-reducing restriction and prolongation operators recently have been developed for the restoration of blur- and noise-contaminated images. This is a particular ill-posed problem. The multilevel methods were found to determine accurate restorations with fairly little computational work. This paper describes noise-reducing multilevel methods for the solution of general linear discrete ill-posed problems.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Baart, M.L.: The use of auto-correlation for pseudo-rank determination in noisy ill-conditioned least-squares problems. IMA J. Numer. Anal. 2, 241–247 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brezinski, C., Redivo-Zaglia, M., Rodriguez, G., Seatzu, S.: Multi-parameter regularization techniques for ill-conditioned linear systems. Numer. Math. 94, 203–228 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for linear systems with applications to regularization. Numer. Algorithms 49, 85–104 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for the regularization of least squares problems. Numer. Algorithms (2009, in press)

  5. Buades, A., Coll, B., Morel, J.M.: The staircasing effect in neighborhood filters and its solution. IEEE Trans. Image Process. 15, 1499–1505 (2006)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Calvetti, D., Lewis, B., Reichel, L.: On the regularizing properties of the GMRES method. Numer. Math. 91, 605–625 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Calvetti, D., Lewis, B., Reichel, L.: On the choice of subspace for iterative methods for linear discrete ill-posed problems. Int. J. Appl. Math. Comput. Sci. 11, 1069–1092 (2001)

    MATH  MathSciNet  Google Scholar 

  9. Calvetti, D., Reichel, L., Zhang, Q.: Conjugate gradient algorithms for symmetric inconsistent linear systems. In: Brown, J.D., Chu, M.T., Ellison, D.C., Plemmons, R.J. (eds.) Proceedings of the Cornelius Lanczos International Centenary Conference, pp. 267–272. SIAM, Philadelphia (1994)

    Google Scholar 

  10. Castellanos, J.L., Gómez, S., Guerra, V.: The triangle method for finding the corner of the L-curve. Appl. Numer. Math. 43, 359–373 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Corsaro, S., Mikula, K., Sarti, A., Sgallari, F.: Semi-implicit covolume method in 3D image segmentation. SIAM J. Sci. Comput. 28, 2248–2265 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Donatelli, M., Serra-Capizzano, S.: On the regularization power of multigrid-type algorithms. SIAM J. Sci. Comput. 27, 2053–2076 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Donatelli, M., Serra-Capizzano, S.: Filter factor analysis of an iterative multilevel regularizing method. Electron. Trans. Numer. Anal. 29, 163–177 (2008)

    MathSciNet  Google Scholar 

  14. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)

    MATH  Google Scholar 

  15. Gulliksson, M., Wedin, P.-Å.: Modifying the QR decomposition to constrained and weighted least-squares. SIAM J. Matrix Anal. 13, 1298–1313 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hanke, M.: Conjugate Gradient Type Methods for Ill-Posed Problems. Longman, Essex (1995)

    MATH  Google Scholar 

  17. Hansen, P.C., Jensen, T.K., Rodriguez, G.: An adaptive pruning algorithm for the discrete L-curve criterion. J. Comput. Appl. Math. 198, 483–492 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM, Philadelphia (1997)

    MATH  Google Scholar 

  19. Hansen, P.C.: Regularization tools, version 4.0 for MATLAB 7.3. Numer. Algorithms 46, 189–294 (2007)

    Article  MATH  Google Scholar 

  20. Morigi, S., Reichel, L., Sgallari, F.: Orthogonal projection regularization operators. Numer. Algorithms 44, 99–114 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Morigi, S., Reichel, L., Sgallari, F.: Cascadic multilevel methods for fast nonsymmetric blurand noise-removal (submitted for publication)

  22. Morigi, S., Reichel, L., Sgallari, F., Shyshkov, A.: Cascadic multiresolution methods for image deblurring. SIAM Imaging J. Sci. 1, 51–74 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Nemirovskii, A.S.: The regularization properties of the adjoint gradient method in ill-posed problems. USSR Comput. Math. Math. Phys. 26(2), 7–16 (1986)

    Article  Google Scholar 

  24. Paige, C.C., Saunders, M.A.: LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. 8, 43–71 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  25. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)

    Article  Google Scholar 

  26. Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first kind. J. ACM 9, 84–97 (1962)

    Article  MATH  Google Scholar 

  27. Reichel, L., Rodriguez, G., Seatzu, S.: Error estimates for large-scale ill-posed problems. Numer. Algorithms (2009, in press)

  28. Reichel, L., Sadok, H.: A new L-curve for ill-posed problems. J. Comput. Appl. Math. 219, 493–508 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Reichel, L., Shyshkov, A.: Cascadic multilevel methods for ill-posed problems. J. Comput. Appl. Math. (2009, in press)

  30. Reichel, L., Ye, Q.: Simple square smoothing regularization operators. Electron. Trans. Numer. Anal. (2009, in press)

  31. Saad, Y.: Iterative Methods for Sparse Linear Systems, vol. 33, 2nd edn. SIAM, Philadelphia (2003)

    Google Scholar 

  32. Scherzer, O.: An iterative multi level algorithm for solving nonlinear ill-posed problems. Numer. Math. 80, 579–600 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  33. Weickert, J., Romeny, B.M.H., Viergever, M.A.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. Image Process. 7, 398–410 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Math.-CIRAM, University of Bologna, Piazza Porta S. Donato 5, 40126, Bologna, Italy

    S. Morigi & F. Sgallari

  2. Department of Mathematical Sciences, Kent State University, Kent, OH, 44242, USA

    L. Reichel

Authors
  1. S. Morigi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Reichel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. Sgallari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Reichel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Morigi, S., Reichel, L. & Sgallari, F. Noise-reducing cascadic multilevel methods for linear discrete ill-posed problems. Numer Algor 53, 1–22 (2010). https://doi.org/10.1007/s11075-009-9282-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-009-9282-3

Keywords

Search

Navigation