BIT Numerical Mathematics

, Volume 58, Issue 3, pp 555–576 | Cite as

Solution methods for linear discrete ill-posed problems for color image restoration

  • A. H. Bentbib
  • M. El Guide
  • K. Jbilou
  • E. Onunwor
  • L. Reichel


This work discusses four algorithms for the solution of linear discrete ill-posed problems with several right-hand side vectors. These algorithms can be applied, for instance, to multi-channel image restoration when the image degradation model is described by a linear system of equations with multiple right-hand sides that are contaminated by errors. Two of the algorithms are block generalizations of the standard Golub–Kahan bidiagonalization method with the block size equal to the number of channels. One algorithm uses standard Golub–Kahan bidiagonalization without restarts for all right-hand sides. These schemes are compared to standard Golub–Kahan bidiagonalization applied to each right-hand side independently. Tikhonov regularization is used to avoid severe error propagation. Numerical examples illustrate the performance of these algorithms. Applications include the restoration of color images.


Golub–Kahan bidiagonalization Block Golub–Kahan bidiagonalization Global Golub–Kahan bidiagonalization Tikhonov regularization Ill-posed problem Multiple right-hand sides Color image restoration 

Mathematics Subject Classification

6510 65F22 



The authors would like to thank Lars Eldén and the referee for carefully reading the manuscript and for comments that improved the presentation. The authors also would like to thank Alessandro Buccini for comments. Research by L.R. is supported in part by NSF Grants DMS-1729509 and DMS-1720259.


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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature, corrected publication May 2018 2018
corrected publication May 2018

Authors and Affiliations

  • A. H. Bentbib
    • 1
  • M. El Guide
    • 1
    • 2
  • K. Jbilou
    • 3
  • E. Onunwor
    • 4
    • 5
  • L. Reichel
    • 4
  1. 1.Laboratoire de Mathématiques Appliquées et InformatiqueFaculté des Sciences et Techniques-GuelizMarrakeshMorocco
  2. 2.Université Mohammed VI PolytechniqueBengeurirMorocco
  3. 3.Université du Littoral Côte d’Opale, L.M.P.A, ULCOCalais-CedexFrance
  4. 4.Department of Mathematical SciencesKent State UniversityKentUSA
  5. 5.Department of MathematicsStark State CollegeNorth CantonUSA

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