Penalized Least-Squares Models For Solving Image Reconstruction Problems

Abstract

The aim of this paper is to improve the performance of the incomplete oblique projections
method(IOP) previously introduced by the authors for solving inconsistent linear systems, when applied
to image reconstruction problems. That method employs incomplete oblique projections onto the set of
solutions of the augmented system Ax ¡ r = b; and converges to a weighted least squares solution of
the system Ax = b. In some applications, in particular for image reconstruction problems, systems are
inconsistent and very often rank-deficient because of the underlying discretized model. It is worthwhile
to point out that the minimum norm solution of the standard least-squares problem is not necessarily
the closest to the seek image. Here, we have considered a penalized least-squares objective function
such that the penalty term incorporates nearest neighbor interactions among adjacent pixels aiming at
smoothing the image. Thus, the oblique incomplete projections algorithm has been modified for solving
penalized problems. The theoretical properties of this new version of the IOP algorithm are analyzed, and
numerical experiences are presented comparing its performance with the one obtained with the original
version applied to the least-squares problem. The tests arise from simulated reconstruction problems of
the SNARK system. They show the new approach improves the quality of the reconstructed images.