Regularized solution of block-banded block Toeplitz systems

Dario Andrea Bini; A. Farusi; G. Fiorentino; Beatrice Meini

doi:10.1117/12.406490

13 November 2000 Regularized solution of block-banded block Toeplitz systems

Dario Andrea Bini, A. Farusi, G. Fiorentino, Beatrice Meini

Proceedings Volume 4116, Advanced Signal Processing Algorithms, Architectures, and Implementations X; (2000) https://doi.org/10.1117/12.406490
Event: International Symposium on Optical Science and Technology, 2000, San Diego, CA, United States

Abstract

Given two n X n Toeplitz matrices T₁ and T₂, and a vector b (epsilon) Rⁿ⁽²), consider the linear system Ax equals b - (eta) , where (eta) (epsilon) Rⁿ⁽²) is an unknown vector representing the noise and A equals T₁ (direct product) T₂. Recovering approximations of x, given A and b, is encountered in image restoration problems. We propose a method for the approximation of the solution x that has good regularization properties. The algorithm is based on a modified version of Newton's iteration for matrix inversion and relies on the concept of approximate displacement rank. We provide a formal description of the regularization properties of Newton's iteration in terms of filters and determine bounds to the number of iterations that guarantee regularization. The method is extended to deal with more general systems where A equals (summation)_{i equals 1}^h T₁⁽ⁱ) (direct product) T₂⁽ⁱ). The cost of computing regularized inverses is O(n log n) operations (ops), the cost of solving the system Ax equals b is O(n² log n) ops. Numerical experiments which show the effectiveness of our algorithm are presented.

Citation Download Citation

Dario Andrea Bini, A. Farusi, G. Fiorentino, and Beatrice Meini "Regularized solution of block-banded block Toeplitz systems", Proc. SPIE 4116, Advanced Signal Processing Algorithms, Architectures, and Implementations X, (13 November 2000); https://doi.org/10.1117/12.406490

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