Extended and constrained diagonal weighting algorithm with application to inverse problems in image reconstruction

The algebraic reconstruction of images in computerized tomography gives rise to large, sparse and ill-conditioned inverse problems, in which the 'effect' (measurement of the attenuation of x-ray intensities after penetration of the analyzed body) is used to compute the 'cause' (the values of the attenuation function inside the body, i.e. the image). Algebraic reconstruction techniques (ART), on both their successive or simultaneous formulation, have been developed since the early 1970s as efficient 'row action methods' for solving the image reconstruction problem in computerized tomography. In this respect, two important development directions were concerned with their extension to the inconsistent case of the reconstruction problem as well as with their combination with constraining strategies, imposed by the particularities of the reconstructed image. In our paper we analyze, from these two points of view, the diagonal weighting (DW) algorithm proposed by Y Censor, D Gordon and R Gordon in 2001 as an improvement of the classical Cimmino's reflection method. In the first part of the paper we introduce general extended and constraining procedures for ART, based on a minimal set of sufficient assumptions that ensure the convergence of the corresponding algorithms. Starting from this general context we then design an extended form of the DW algorithm together with a constraining procedure for which we prove convergence under appropriate assumptions. Numerical experiments are presented on two phantoms widely used in the literature.