Symmetrization Techniques in Image Deblurring

TL;DR

This paper introduces novel preconditioning techniques that significantly improve the efficiency of iterative regularization methods for image deblurring across various boundary conditions and PSFs, supported by theoretical analysis and extensive experiments.

Contribution

It proposes new symmetrization preconditioning methods for iterative regularization in image deblurring, enhancing convergence and performance for diverse boundary conditions and PSFs.

Findings

Anti-identity preconditioner improves MINRES for zero boundary conditions.

Circulant preconditioners accelerate GMRES and flexible Krylov methods.

Numerical experiments confirm the effectiveness of the proposed techniques.

Abstract

This paper presents a couple of preconditioning techniques that can be used to enhance the performance of iterative regularization methods applied to image deblurring problems with a variety of point spread functions (PSFs) and boundary conditions. More precisely, 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, 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. The results of many numerical experiments are reported to show the effectiveness of the new preconditioning techniques, including when considering the deblurring of sparse images.