A note on Kaczmarz algorithm with remotest set control sequence

TL;DR

This paper analyzes a variant of the Kaczmarz algorithm that selects the most distant residual for projection, proving that each row is chosen at least once in certain systems relevant to image reconstruction.

Contribution

It introduces and analyzes a remotest set control sequence for the Kaczmarz algorithm, ensuring each row is selected at least once in specific underdetermined systems.

Findings

Each row index is selected at least once during iterations.

The method is applicable to algebraic reconstruction in computerized tomography.

The selection procedure improves understanding of convergence properties.

Abstract

In this paper we analyse the Kaczmarz projection algorithm with remotest set control of projection indices. According to this procedure, in each iteration the projection index is one which gives the maximal absolute value of the corresponding residual. We prove that for underdetermined full row rank systems and under some assumptions valid for problems arising in algebraic reconstruction of images in computerized tomography, this selection procedure has the property that each row index is selected at least once during the Kaczmarz algorithm iterations.