- Research Article
- Open Access
Regularizing Inverse Preconditioners for Symmetric Band Toeplitz Matrices
EURASIP Journal on Advances in Signal Processing volume 2007, Article number: 085606 (2007)
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.
Hanke M, Nagy J: Inverse Toeplitz preconditioners for ill-posed problems. Linear Algebra and Its Applications 1998,284(1–3):137-156.
Hanke M: Conjugate Gradient Type Methods for Ill-Posed Problems, Pitman Research Notes in Mathematics. Longman, Harlow, UK; 1995.
Hanke M: Iterative regularization techniques in image restoration. In Mathematical Methods in Inverse Problems for Partial Differential Equations. Springer, New York, NY, USA; 1998.
Hansen PE: Rank-Deficient and Discrete Ill-Posed Problems, SIAM Monographs on Mathematical Modeling and Computation. SIAM, Philadelphia, Pa, USA; 1998.
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.
Jin X-Q: Developments and Applications of Block Toeplitz Iterative Solvers. Kluwer Academic Publishers, Dordrecht, The Netherlands; Science Press, Beijing, China; 2002.
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
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
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
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
Chan RH, Ng K-P: Toeplitz preconditioners for Hermitian Toeplitz systems. Linear Algebra and Its Applications 1993, 190: 181–208.
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
Grenander U, Szegö G: Toeplitz Forms and Their Applications. 2nd edition. Chelsea, New York, NY, USA; 1984.
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.
Parlett BN: The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, NJ, USA; 1980.
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.
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.
Tyrtyshnikov EE: Optimal and superoptimal circulant preconditioners. SIAM Journal on Matrix Analysis and Applications 1992,13(2):459-473. 10.1137/0613030
Golub GH, Van Loan C: Matrix Computation. Academic Press, New York, NY, USA; 1981.
Dahlquist G, Björck A: Numerical Methods. Prentice-Hall, Englewood Cliffs, NJ, USA; 1974.
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
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
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
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
NRL Monterey Marine Meteorology Division (Code 7500) https://doi.org/www.nrlmry.navy.mil/sat_products.html
About this article
Cite this article
Favati, P., Lotti, G. & Menchi, O. Regularizing Inverse Preconditioners for Symmetric Band Toeplitz Matrices. EURASIP J. Adv. Signal Process. 2007, 085606 (2007) doi:10.1155/2007/85606
- Information Technology
- Linear System
- Iterative Method
- Quantum Information
- Regularization Method