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Regularizing Inverse Preconditioners for Symmetric Band Toeplitz Matrices

Abstract

Image restoration is a widely studied discrete ill-posed problem. Among the many regularization methods used for treating the problem, iterative methods have been shown to be effective. In this paper, we consider the case of a blurring function defined by space invariant and band-limited PSF, modeled by a linear system that has a band block Toeplitz structure with band Toeplitz blocks. In order to reduce the number of iterations required to obtain acceptable reconstructions, in 13 an inverse Toeplitz preconditioner for problems with a Toeplitz structure was proposed. The cost per iteration is of operations, where is the pixel number of the 2D image. In this paper, we propose inverse preconditioners with a band Toeplitz structure, which lower the cost to and in experiments showed the same speed of convergence and reconstruction efficiency as the inverse Toeplitz preconditioner.

References

  1. 1.

    Hanke M, Nagy J: Inverse Toeplitz preconditioners for ill-posed problems. Linear Algebra and Its Applications 1998,284(1–3):137-156.

    MathSciNet  Article  Google Scholar 

  2. 2.

    Hanke M: Conjugate Gradient Type Methods for Ill-Posed Problems, Pitman Research Notes in Mathematics. Longman, Harlow, UK; 1995.

    Google Scholar 

  3. 3.

    Hanke M: Iterative regularization techniques in image restoration. In Mathematical Methods in Inverse Problems for Partial Differential Equations. Springer, New York, NY, USA; 1998.

    Google Scholar 

  4. 4.

    Hansen PE: Rank-Deficient and Discrete Ill-Posed Problems, SIAM Monographs on Mathematical Modeling and Computation. SIAM, Philadelphia, Pa, USA; 1998.

    Google Scholar 

  5. 5.

    Hanke M, Nagy J, Plemmons R: Preconditioned iterative regularization for ill-posed problems. In Numerical Linear Algebra and Scientific Computing. Edited by: Reichel L, Ruttan A, Varga RS. de Gruyter, Berlin, Germany; 1993:141-163.

    Google Scholar 

  6. 6.

    Jin X-Q: Developments and Applications of Block Toeplitz Iterative Solvers. Kluwer Academic Publishers, Dordrecht, The Netherlands; Science Press, Beijing, China; 2002.

    Google Scholar 

  7. 7.

    Chan RH, Tang P: Fast band-Toeplitz preconditioner for Hermitian Toeplitz systems. SIAM Journal on Scientific Computing 1994,15(1):164-171. 10.1137/0915011

    MathSciNet  Article  Google Scholar 

  8. 8.

    Jin X-Q: Band Toeplitz preconditioners for block Toeplitz systems. Journal of Computational and Applied Mathematics 1996,70(2):225-230. 10.1016/0377-0427(95)00205-7

    MathSciNet  Article  Google Scholar 

  9. 9.

    Serra Capizzano S: Optimal, quasi-optimal and superlinear band-Toeplitz preconditioners for asymptotically ill-conditioned positive definite Toeplitz systems. Mathematics of Computation 1997,66(218):651-665. 10.1090/S0025-5718-97-00833-8

    MathSciNet  Article  Google Scholar 

  10. 10.

    Favati P, Lotti G, Menchi O: Preconditioners based on fit techniques for the iterative regularization in the image deconvolution problem. BIT Numerical Mathematics 2005,45(1):15-35. 10.1007/s10543-005-2639-7

    MathSciNet  Article  Google Scholar 

  11. 11.

    Chan RH, Ng K-P: Toeplitz preconditioners for Hermitian Toeplitz systems. Linear Algebra and Its Applications 1993, 190: 181–208.

    MathSciNet  Article  Google Scholar 

  12. 12.

    Hanke M, Nagy J: Restoration of atmospherically blurred images by symmetric indefinite conjugate gradient techniques. Inverse Problems 1996,12(2):157-173. 10.1088/0266-5611/12/2/004

    MathSciNet  Article  Google Scholar 

  13. 13.

    Grenander U, Szegö G: Toeplitz Forms and Their Applications. 2nd edition. Chelsea, New York, NY, USA; 1984.

    Google Scholar 

  14. 14.

    Tilli P: Asymptotic spectral distribution of Toeplitz-related matrices. In Fast Reliable Algorithms for Matrices with Structure. Edited by: Kailath T, Sayed AH. SIAM, Philadelphia, Pa, USA; 1999:153-187.

    Google Scholar 

  15. 15.

    Parlett BN: The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, NJ, USA; 1980.

    Google Scholar 

  16. 16.

    Favati P, Lotti G, Menchi O: A polynomial fit preconditioner for band Toeplitz matrices in image reconstruction. Linear Algebra and Its Applications 2002,346(1–3):177-197.

    MathSciNet  Article  Google Scholar 

  17. 17.

    Lei S-L, Kou K-I, Jin X-Q: Preconditioners for ill-conditioned block Toeplitz systems with application in image restoration. East-West Journal of Numerical Mathematics 1999,7(3):175-185.

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Tyrtyshnikov EE: Optimal and superoptimal circulant preconditioners. SIAM Journal on Matrix Analysis and Applications 1992,13(2):459-473. 10.1137/0613030

    MathSciNet  Article  Google Scholar 

  19. 19.

    Golub GH, Van Loan C: Matrix Computation. Academic Press, New York, NY, USA; 1981.

    Google Scholar 

  20. 20.

    Dahlquist G, Björck A: Numerical Methods. Prentice-Hall, Englewood Cliffs, NJ, USA; 1974.

    Google Scholar 

  21. 21.

    Bini DA, Favati P, Menchi O: A family of modified regularizing circulant preconditioners for two-levels Toeplitz systems. Computers & Mathematics with Applications 2004,48(5-6):755-768. 10.1016/j.camwa.2004.03.006

    MathSciNet  Article  Google Scholar 

  22. 22.

    Di Benedetto F, Serra Capizzano S: A note on the superoptimal matrix algebra operators. Linear and Multilinear Algebra 2002,50(4):343-372. 10.1080/0308108021000049301

    MathSciNet  Article  Google Scholar 

  23. 23.

    Di Benedetto F, Estatico C, Serra Capizzano S: Superoptimal preconditioned conjugate gradient iteration for image deblurring. SIAM Journal of Scientific Computing 2005,26(3):1012-1035. 10.1137/S1064827503421653

    MathSciNet  Article  Google Scholar 

  24. 24.

    Lee KP, Nagy J, Perrone L: Iterative methods for image restoration: a Matlab object oriented approach. 2002.https://doi.org/www.mathcs.emory.edu/~nagy/RestoreTools

    Google Scholar 

  25. 25.

    NRL Monterey Marine Meteorology Division (Code 7500) https://doi.org/www.nrlmry.navy.mil/sat_products.html

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Correspondence to P. Favati.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://doi.org/creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Favati, P., Lotti, G. & Menchi, O. Regularizing Inverse Preconditioners for Symmetric Band Toeplitz Matrices. EURASIP J. Adv. Signal Process. 2007, 085606 (2007). https://doi.org/10.1155/2007/85606

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Keywords

  • Information Technology
  • Linear System
  • Iterative Method
  • Quantum Information
  • Regularization Method