Volume 59, pp. 157-178, 2023.

Symmetrization techniques in image deblurring

Marco Donatelli, Paola Ferrari, and Silvia Gazzola

Abstract

This paper presents some preconditioning techniques that enhance the performance of iterative regularization methods applied to image deblurring problems determined by a wide variety of point spread functions (PSFs) and boundary conditions. We first consider the anti-identity preconditioner, which symmetrizes the coefficient matrix associated to problems with zero boundary conditions, allowing the use of MINRES as a regularization method. When considering more sophisticated boundary conditions and strongly nonsymmetric PSFs, we show that the anti-identity preconditioner improves the performance of GMRES. We then consider both stationary and iteration-dependent regularizing circulant preconditioners that, applied in connection with the anti-identity matrix and both standard and flexible Krylov subspaces, speed up the iterations. A theoretical result about the clustering of the eigenvalues of the preconditioned matrices is proved in a special case. Extensive numerical experiments show the effectiveness of the new preconditioning techniques, including when the deblurring of sparse images is considered.

Full Text (PDF) [5.5 MB], BibTeX

Key words

inverse problems, regularization, preconditioning, Toeplitz matrices, Krylov subspace methods

AMS subject classifications

65F08, 65F10, 65F22

Links to the cited ETNA articles

[2]	Vol. 53 (2020), pp. 113-216 Giovanni Barbarino, Carlo Garoni, and Stefano Serra-Capizzano: Block generalized locally Toeplitz sequences: theory and applications in the multidimensional case

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