Incomplete oblique projections method for solving regularized least-squares problems in image reconstruction

In this paper we improve on the incomplete oblique projections (IOP) method introduced previously by the authors for solving inconsistent linear systems, when applied to image reconstruction problems. That method uses IOP 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 image reconstruction problems, systems are usually inconsistent and very often rank‐deficient because of the underlying discretized model. Here we have considered a regularized least‐squares objective function that can be used in many ways such as incorporating blobs or nearest‐neighbor interactions among adjacent pixels, aiming at smoothing the image. Thus, the oblique incomplete projections algorithm has been modified for solving this regularized model. The theoretical properties of the new algorithm are analyzed and numerical experiments are presented showing that the new approach improves the quality of the reconstructed images.