# Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector

## TL;DR

This paper addresses convergence issues in algebraic iterative reconstruction methods with unmatched projector/backprojector pairs by proposing a shifted algorithm that guarantees convergence under certain conditions, with practical implementation guidance.

## Contribution

It introduces a shifted algorithm for unmatched algebraic reconstruction that ensures convergence and provides eigenvalue estimation techniques for parameter selection.

## Key findings

- The shifted algorithm guarantees convergence under specific conditions.
- Perturbation bounds for the fixed point are established.
- Numerical tests demonstrate improved convergence in computed tomography.

## Abstract

We consider algebraic iterative reconstruction methods with applications in image reconstruction. In particular, we are concerned with methods based on an unmatched projector/backprojector pair; i.e., the backprojector is not the exact adjoint or transpose of the forward projector. Such situations are common in large-scale computed tomography, and we consider the common situation where the method does not converge due to the nonsymmetry of the iteration matrix. We propose a modified algorithm that incorporates a small shift parameter, and we give the conditions that guarantee convergence of this method to a fixed point of a slightly perturbed problem. We also give perturbation bounds for this fixed point. Moreover, we discuss how to use Krylov subspace methods to efficiently estimate the leftmost eigenvalue of a certain matrix to select a proper shift parameter. The modified algorithm is illustrated with test problems from computed tomography.