Error Correction for Discrete Tomography

TL;DR

This paper addresses error correction in discrete tomography, demonstrating that fewer than half of the line sums can be corrected and establishing this as the optimal bound.

Contribution

It introduces a method to correct errors in line sums in discrete tomography and proves the correction bound is optimal.

Findings

Less than d/2 errors can be corrected in line sums.

The correction bound of less than d/2 errors is proven to be optimal.

The method extends previous reconstruction techniques to handle data errors.

Abstract

Discrete tomography focuses on the reconstruction of functions $f: A \to \mathbb{R}$ from their line sums in a finite number $d$ of directions, where $A$ is a finite subset of $\mathbb{Z}^2$. Consequently, the techniques of discrete tomography often find application in areas where only a small number of projections are available. In 1978 M.B. Katz gave a necessary and sufficient condition for the uniqueness of the solution. Since then, several reconstruction methods have been introduced. Recently Pagani and Tijdeman developed a fast method to reconstruct $f$ if it is uniquely determined. Subsequently Ceko, Pagani and Tijdeman extended the method to the reconstruction of a function with the same line sums of $f$ in the general case. Up to here we assumed that the line sums are exact. In this paper we investigate the case where a small number of line sums are incorrect as may happen when discrete tomography is applied for data storage or transmission. We show how less than $d/2$ errors can be corrected and that this bound is the best possible.