A Fast Alternating Minimization Algorithm for Total Variation Deblurring Without Boundary Artifacts

TL;DR

This paper introduces a continuous alternating minimization algorithm for total variation image deblurring that effectively handles boundary artifacts, providing higher quality restorations with comparable computational efficiency.

Contribution

It develops a continuous version of the alternating minimization algorithm for TV deblurring, with proven convergence and strategies for boundary artifact mitigation.

Findings

Higher quality image restoration compared to existing methods

Comparable running times to fast TV deconvolution algorithms

Effective boundary artifact handling strategies

Abstract

Recently, a fast alternating minimization algorithm for total variation image deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang [{\em SIAM J. Imaging Sci.}, 1 (2008), pp. 248--272]. The method in a nutshell consists of a discrete Fourier transform-based alternating minimization algorithm with periodic boundary conditions and in which two fast Fourier transforms (FFTs) are required per iteration. In this paper, we propose an alternating minimization algorithm for the continuous version of the total variation image deblurring problem. We establish convergence of the proposed continuous alternating minimization algorithm. The continuous setting is very useful to have a unifying representation of the algorithm, independently of the discrete approximation of the deconvolution problem, in particular concerning the strategies for dealing with boundary artifacts. Indeed, an accurate restoration of blurred and noisy images requires a proper treatment of the boundary. A discrete version of our continuous alternating minimization algorithm is obtained following two different strategies: the imposition of appropriate boundary conditions and the enlargement of the domain. The first one is computationally useful in the case of a symmetric blur, while the second one can be efficiently applied for a nonsymmetric blur. Numerical tests show that our algorithm generates higher quality images in comparable running times with respect to the Fast Total Variation deconvolution algorithm.