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The aim of this paper is to extend the applicability of an algorithm for solving inconsistent linear systems to the rank-deficient case, by employing incomplete projections onto the set of solutions of the augmented system Ax − r = b. The extended algorithm converges to the unique minimal norm solution of the least squares solutions. For that purpose, incomplete oblique projections are used, defined by means of matrices that penalize the norm of the residuals. The theoretical properties of the new algorithm are analyzed, and numerical experiences are presented comparing its performance with some well-known projection methods.

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.

The Projected Aggregation Methods (PAM) for solving linear systems of equalities and/or inequalities, generate a new iterate x k+1 by projecting the current point x k onto a separating hyperplane generated by a given linear combination of the original hyperplanes or halfspaces. In Scolnik et al. (2001Scolnik et al. ( , 2002a and Echebest et al. (2004) acceleration schemes for solving systems of linear equations and inequalities respectively were introduced, within a PAM like framework. In this paper we apply those schemes in an algorithm based on oblique projections reflecting the sparsity of the matrix of the linear system to be solved. We present the corresponding theoretical convergence results which are a generalization of those given in Echebest et al. (2004). We also present the numerical results obtained applying the new scheme to two algorithms introduced by García-Palomares and González-Castaño (1998) and also the comparison of its efficiency with that of Censor and Elfving (2002). Keywords: projected aggregation methods, exact projection, incomplete projections, oblique projectionsThe class of convex feasibility problems (CFP) consisting in finding an element of a non-empty closed C convex set which is a subset of n , andis the intersection of a family of closed convex subsets C i , i = 1, 2, . . . , m of the ndimensional Euclidean space, have many applications in various fields of science and technology, particularly in problems of image reconstruction from projections (Censor, 1988;Herman and Meyer, 1993). Solving systems of linear equalities and/or inequalities is one of them. A common approach to such problems is to use projection algorithms *