# A twin error gauge for Kaczmarz's iterations

## TL;DR

This paper introduces two novel algebraic reconstruction methods based on Kaczmarz's iterations that utilize a twin error gauge to improve regularized solutions in noisy tomography, outperforming standard stopping rules.

## Contribution

The paper presents a new error gauge using paired up- and down-sweep Kaczmarz methods, enabling better stopping criteria and improved reconstruction accuracy.

## Key findings

- Lower two-norm error in most test cases compared to standard methods.
- Computational cost is slightly less than standard Kaczmarz with statistical stopping.
- Effective regularization in noisy tomography problems.

## Abstract

We propose two new algebraic reconstruction techniques based on Kaczmarz's method that produce a regularized solution to noisy tomography problems. Tomography problems exhibit semi-convergence when iterative methods are employed, and the aim is therefore to stop near the semi-convergence point. Our approach is based on an error gauge that is constructed by pairing standard down-sweep Kaczmarz's method with its up-sweep version; we stop the iterations when this error gauge is minimal. The reconstructions of the new methods differ from standard Kaczmarz iterates in that our final result is the average of the stopped up- and down-sweeps. Even when Kaczmarz's method is supplied with an oracle that provides the exact error -- and is thereby able to stop at the best possible iterate -- our methods have a lower two-norm error in the vast majority of our test cases. In terms of computational cost, our methods are a little cheaper than standard Kaczmarz equipped with a statistical stopping rule.