A Preconditioned Version of a Nested Primal-Dual Algorithm for Image Deblurring

TL;DR

This paper introduces a preconditioned variant of a nested primal-dual algorithm for image deblurring, demonstrating improved computational efficiency and comparable solution quality through theoretical convergence proofs and numerical experiments.

Contribution

It reinterprets variable metric strategies as preconditioning methods and develops an inexact left-preconditioned proximal gradient algorithm with proven convergence for image deblurring.

Findings

Left preconditioning reduces CPU time compared to right preconditioning.

Both preconditioning methods require similar iteration counts to reach convergence.

Proposed non-stationary preconditioner sequences enable fast, stable convergence.

Abstract

Variational models for image deblurring problems typically consist of a smooth term and a potentially non-smooth convex term. A common approach to solving these problems is using proximal gradient methods. To accelerate the convergence of these first-order iterative algorithms, strategies such as variable metric methods have been introduced in the literature. In this paper, we prove that, for image deblurring problems, the variable metric strategy can be reinterpreted as a right preconditioning method. Consequently, we explore an inexact left-preconditioned version of the same proximal gradient method. We prove the convergence of the new iteration to the minimum of a variational model where the norm of the data fidelity term depends on the preconditioner. The numerical results show that left and right preconditioning are comparable in terms of the number of iterations required to reach a prescribed tolerance, but left preconditioning needs much less CPU time, as it involves fewer evaluations of the preconditioner matrix compared to right preconditioning. The quality of the computed solutions with left and right preconditioning are comparable. Finally, we propose some non-stationary sequences of preconditioners that allow for fast and stable convergence to the solution of the variational problem with the classical $\ell^2$--norm on the fidelity term.