Implicit regularization of the incomplete oblique projections method

Abstract

AbstractThe 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. Many tomographic image reconstruction problems are such that the limitation of the range of rays makes the model underdetermined, the discretized linear system is rank‐deficient, the nullspace is non‐trivial, and the minimal norm least squares solution may be far away from the true image. In a previous paper, we have added a quadratic term reflecting neighboring pixel information to the standard least squares model for improving the quality of the reconstructed images. In this paper we replace the quadratic function by a more general regularizing function avoiding the modification of the original system. The key idea is to perform a joint optimization of the norm of the residual and of the regularizing function in each iteration. The theoretical properties of this new algorithm are analyzed, and numerical experiments are presented comparing its performance with other well‐known methods. They show that the new approach improves the quality of the reconstructed images.