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A General Framework for Nonlinear Regularized Krylov-Based Image Restoration

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Computational Modeling of Objects Presented in Images. Fundamentals, Methods, and Applications (CompIMAGE 2014)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 8641))

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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.

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References

  1. Björck, Å.: A bidiagonalization algorithm for solving large and sparse ill-posed systems of linear equations. BIT Numer. Math. 28, 659–670 (1988)

    Article  MATH  Google Scholar 

  2. Calvetti, D., Reichel, L.: Tikhonov regularization of large linear problems. BIT Numer. Math. 43, 263–283 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Eldén, L.: A weighted pseudoinverse, generalized singular values, and constrained least squares problems. BIT Numer. Math. 22, 487–502 (1982)

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  7. Hochstenbach, M.E., Reichel, L., Yu, X.: A Golub–Kahan-type reduction method for matrix pairs (submitted for publication)

    Google Scholar 

  8. Lampe, J., Reichel, L., Voss, H.: Large-scale Tikhonov regularization via reduction by orthogonal projection. Linear Algebra Appl. 436, 2845–2865 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Reichel, L., Sgallari, F., Ye, Q.: Tikhonov regularization based on generalized Krylov subspace methods. Appl. Numer. Math. 62, 1215–1228 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bologna, Bologna, Italy

    Serena Morigi & Fiorella Sgallari

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

    Lothar Reichel

Authors
  1. Serena Morigi
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  2. Lothar Reichel
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  3. Fiorella Sgallari
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Editor information

Editors and Affiliations

  1. Department of Mechanical Engineering, Carnegie Mellon University, 5000 Forbes Avenue, 15213, Pittsburgh, PA, USA

    Yongjie Jessica Zhang

  2. Faculdade de Engenharia, Departamento de Engenharia Mecânica, Universidade do Porto, Rua Dr. Roberto Frias, S/N, 4200-465, Porto, Portugal

    João Manuel R. S. Tavares

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© 2014 Springer International Publishing Switzerland

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Morigi, S., Reichel, L., Sgallari, F. (2014). A General Framework for Nonlinear Regularized Krylov-Based Image Restoration. In: Zhang, Y.J., Tavares, J.M.R.S. (eds) Computational Modeling of Objects Presented in Images. Fundamentals, Methods, and Applications. CompIMAGE 2014. Lecture Notes in Computer Science, vol 8641. Springer, Cham. https://doi.org/10.1007/978-3-319-09994-1_27

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  • DOI: https://doi.org/10.1007/978-3-319-09994-1_27

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-09993-4

  • Online ISBN: 978-3-319-09994-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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