BIT Numerical Mathematics

, Volume 53, Issue 2, pp 459–473 | Cite as

Data based regularization for discrete deconvolution problems



We focus on the solution of discrete deconvolution problems to recover the original information from blurred signals in the presence of Gaussian white noise more accurately. For a certain class of blur operators and signals we develop a diagonal preconditioner to improve the reconstruction quality, both for direct and iterative regularization methods. In this respect, we incorporate the variation of the signal data during the construction of the preconditioner. Embedding this method in an outer iteration may yield further improvement of the solution. Numerical examples demonstrate the effect of the presented approach.


Tikhonov-Phillips TSVD CGLS Ill-posed inverse problems 

Mathematics Subject Classification (2010)

65F22 65F08 65R30 


  1. 1.
    Björck, Å.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996) MATHCrossRefGoogle Scholar
  2. 2.
    Calvetti, D., Lewis, B., Reichel, L.: GMRES-type methods for inconsistent systems. Linear Algebra Appl. 316, 157–169 (2000) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996) MATHCrossRefGoogle Scholar
  4. 4.
    Hanke, M.: Conjugate Gradient Type Methods for Ill-Posed Problems. Longman Scientific & Technical, Harlow, Essex (1995) MATHGoogle Scholar
  5. 5.
    Hanke, M., Hansen, P.C.: Regularization methods for large-scale problems. Surv. Math. Ind. 3, 253–315 (1993) MathSciNetMATHGoogle Scholar
  6. 6.
    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 and Scientific Computing, pp. 141–163. de Gruyter, Berlin (1993) Google Scholar
  7. 7.
    Hansen, P.C.: The truncated SVD as a method for regularization. BIT Numer. Math. 27(4), 534–553 (1987) MATHCrossRefGoogle Scholar
  8. 8.
    Hansen, P.C.: Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems. Numer. Algorithms 6(1–2), 1–35 (1994) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Hansen, P.C.: Discrete Inverse Problems: Insight and Algorithms, 1st edn. SIAM, Philadelphia (2010) MATHCrossRefGoogle Scholar
  10. 10.
    Huckle, T., Kallischko, A.: Frobenius norm minimization and probing for preconditioning. Int. J. Comput. Math. 84, 1225–1248 (2007) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Huckle, T., Sedlacek, M.: Smoothing and regularization with modified sparse approximate inverses. J. Electr. Comput. Eng. 2010, 1–16 (2010). Iterative Signal Processing in Communications. MathSciNetCrossRefGoogle Scholar
  12. 12.
    Huckle, T., Sedlacek, M.: Tikhonov-Phillips regularization with operator dependent seminorms. Numer. Algorithms 60(2), 339–353 (2012) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Huckle, T., Sedlacek, M.: Data based regularization matrices for the Tikhonov-Phillips regularization. In: Proceedings of the 83rd Annual Meeting of the International Association of Applied Mathematics and Mechanics. PAMM Proc. Appl. Math. Mech., vol. 12, pp. 643–644 (2012) Google Scholar
  14. 14.
    Jensen, T.K., Hansen, P.C.: Iterative regularization with minimum-residual methods. BIT Numer. Math. 47, 103–120 (2007) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    MATLAB: version 7.11.0 (R2010b), The MathWorks Inc. (2010) Google Scholar
  16. 16.
    Noschese, S., Reichel, L.: Inverse problems for regularization matrices. Numer. Algorithms 60(4), 531–544 (2012) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Paige, C.C., Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12(4), 617–629 (1975) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first kind. J. Assoc. Comput. Mach. 9(1), 84–97 (1962) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Tikhonov, A.N.: Solution of incorrectly formulated problems and regularization method. Sov. Math. Dokl. 4, 1035–1038 (1963). English translation of Dokl. Akad. Nauk. SSSR 151, 501–504 (1963) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Fakultät für InformatikTechnische Universität MünchenGarchingGermany

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