GMRES Methods for Tomographic Reconstruction with an Unmatched Back Projector

TL;DR

This paper introduces AB- and BA-GMRES algorithms for tomographic reconstruction that effectively handle unmatched forward and back projectors, offering a simple, parameter-free solution suitable for large-scale, noisy CT problems.

Contribution

The paper develops and analyzes AB- and BA-GMRES algorithms for unmatched projector pairs, demonstrating their equivalence to LSQR/LSMR and their effectiveness in large-scale CT reconstructions.

Findings

Algorithms exhibit semi-convergence similar to LSQR/LSMR.

They are simple to implement and parameter-free.

Effective for large-scale, noisy CT reconstruction problems.

Abstract

Unmatched pairs of forward and back projectors are common in X-ray CT computations for large-scale problems; they are caused by the need for fast algorithms that best utilize the computer hardware, and it is an interesting and challenging task to develop fast and easy-to-use algorithms for these cases. Our approach is to use preconditioned GMRES, in the form of the AB- and BA-GMRES algorithms, to handle the unmatched normal equations associated with an unmatched pair. These algorithms are simple to implement, they rely only on computations with the available forward and back projectors, and they do not require the tuning of any algorithm parameters. We show that these algorithms are equivalent to well-known LSQR and LSMR algorithms in the case of a matched projector. Our numerical experiments demonstrate that AB- and BA-GMRES exhibit a desired semi-convergence behavior that is comparable with LSQR/LSMR and that standard stopping rules work well. Hence, AB- and BA-GMRES are suited for large-scale CT reconstruction problems with noisy data and unmatched projector pairs.