Extended and Constrained Cimmino-type Algorithms with Applications in Tomographic Image Reconstruction

Abstract

In the first part of this paper we propose an extension of Cimmino's reflections algorithm to inconsistent least squares problems. For proving convergence of the extended algorithm we first make a convergence analysis of the classical method that reveals other interesting aspects which don't appear in the convergence analysis made in the original paper by G. Cimmino. In the second part of the paper we introduce in both original and extended Cimmino algorithms constraining procedures and prove convergence of mixed algorithms in both cases to constrained classical or least squares solutions, respectively. In the third part of the paper we present numerical experiments on some algebraic image reconstruction models, with particular attention to tomographic particle image reconstruction, which illustrate the performance of both extended and constrained algorithms.