Unmatched Projector/Backprojector Pairs: Perturbation and Convergence Analysis

In tomographic reconstruction problems it is not uncommon that there are errors in the implementation of the forward projector and/or the backprojector, and hence we encounter a so-called unmatched projektor/backprojector pair. Consequently, the matrices that represent the two projectors are not each other's transpose. Surprisingly, the influence of such errors in algebraic iterative reconstruction methods has received little attention in the literature. The goal of this paper is to perform a rigorous first-order perturbation analysis of the minimization problems underlying the algebraic methods in order to understand the role played by the nonmatch of the matrices. We also study the convergence properties of linear stationary iterations based on unmatched matrix pairs, leading to insight into the behavior of some important row-and column-oriented algebraic iterative methods. We conclude with numerical examples that illustrate the perturbation and convergence results.