Advertisement

A General Framework for Nonlinear Regularized Krylov-Based Image Restoration

  • Serena Morigi
  • Lothar Reichel
  • Fiorella Sgallari
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8641)

Abstract

This paper introduces a new approach to computing an approximate solution of Tikhonov-regularized large-scale ill-posed problems with a general nonlinear regularization operator. The iterative method applies a sequence of projections onto generalized Krylov subspaces using a semi-implicit approach to deal with the nonlinearity in the regularization term. A suitable value of the regularization parameter is determined by the discrepancy principle. Computed examples illustrate the performance of the method applied to the restoration of blurred and noisy images.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Björck, Å.: A bidiagonalization algorithm for solving large and sparse ill-posed systems of linear equations. BIT Numer. Math. 28, 659–670 (1988)CrossRefzbMATHGoogle Scholar
  2. 2.
    Calvetti, D., Reichel, L.: Tikhonov regularization of large linear problems. BIT Numer. Math. 43, 263–283 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Tikhonov regularization and the L-curve for large, discrete ill-posed problems. J. Comput. Appl. Math. 123, 423–446 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Eldén, L.: A weighted pseudoinverse, generalized singular values, and constrained least squares problems. BIT Numer. Math. 22, 487–502 (1982)CrossRefzbMATHGoogle Scholar
  5. 5.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)CrossRefzbMATHGoogle Scholar
  6. 6.
    Hochstenbach, M.E., Reichel, L.: An iterative method for Tikhonov regularization with a general linear regularization operator. J. Integral Equations Appl. 22, 463–480 (2010)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Hochstenbach, M.E., Reichel, L., Yu, X.: A Golub–Kahan-type reduction method for matrix pairs (submitted for publication)Google Scholar
  8. 8.
    Lampe, J., Reichel, L., Voss, H.: Large-scale Tikhonov regularization via reduction by orthogonal projection. Linear Algebra Appl. 436, 2845–2865 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Reichel, L., Sgallari, F., Ye, Q.: Tikhonov regularization based on generalized Krylov subspace methods. Appl. Numer. Math. 62, 1215–1228 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)CrossRefzbMATHGoogle Scholar
  11. 11.
    Weickert, J., Romeny, B.M.H., Viergever, M.A.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. Image Processing 7, 398–410 (1998)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Serena Morigi
    • 1
  • Lothar Reichel
    • 2
  • Fiorella Sgallari
    • 1
  1. 1.Department of MathematicsUniversity of BolognaBolognaItaly
  2. 2.Department of Mathematical SciencesKent State UniversityKentUSA

Personalised recommendations