Fast Preconditioners for Total Variation Deblurring with Anti-Reflective Boundary Conditions

TL;DR

This paper introduces fast preconditioning strategies for Total Variation deblurring that leverage anti-reflective boundary conditions and trigonometric transforms, improving both accuracy and computational efficiency.

Contribution

It combines precise boundary conditions with TV regularization and proposes new preconditioners based on fast transforms, enhancing deblurring performance.

Findings

Anti-reflective BCs reduce analytical error.

Preconditioners improve convergence speed.

Overall method lowers computational cost.

Abstract

In recent works several authors have proposed the use of precise boundary conditions (BCs) for blurring models and they proved that the resulting choice (Neumann or reflective, anti-reflective) leads to fast algorithms both for deblurring and for detecting the regularization parameters in presence of noise. When considering a symmetric point spread function, the crucial fact is that such BCs are related to fast trigonometric transforms. In this paper we combine the use of precise BCs with the Total Variation (TV) approach in order to preserve the jumps of the given signal (edges of the given image) as much as possible. We consider a classic fixed point method with a preconditioned Krylov method (usually the conjugate gradient method) for the inner iteration. Based on fast trigonometric transforms, we propose some preconditioning strategies which are suitable for reflective and anti-reflective BCs. A theoretical analysis motivates the choice of our preconditioners and an extensive numerical experimentation is reported and critically discussed. The latter shows that the TV regularization with anti-reflective BCs implies not only a reduced analytical error, but also a lower computational cost of the whole restoration procedure over the other BCs.