Algebraic iterative methods are popular for CT reconstruction due to their ability to adapt to the geometry of the CT scanner and the measurements, and to handle limited-angle and limited-data problems.  Well-known examples are Kaczmarz's method and variants of Landweber and Cimmino iterations.  These methods, and their block extensions, utilize projection and backprojection operations in each iterative step.  Both operations are defined by the geometry and the physics of the problem, and when discretized the projection is represented by a matrix A while the backprojection is, in principle, represented by AT (the transpose of A).

The particular discretization methods used to obtain the projection and backprojection depend on the application and, to some extent, also on traditions in the specific application communities.  Moreover, it is often the case that the software uses different discretization methods for the projection and the backprojection, e.g., due to considerations about the most efficient use of computer hardware.

Consequently, in all these circumstances the matrix that represents the backprojection is not equal to AT, a situation referred to as an unmatched projector/backprojector pair.  It is therefore relevant to study the influence of such an unmatched pair on the reconstruction problem, as well as their influence on the convergence properties of the algebraic iterative methods applied to the unmatched problem.

We present a perturbation analysis of the minimization problems underlying the algebraic iterative methods, in order to understand the role played by the non-match 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.

This is a joint work with Tommy Elfving, Linköping University.