Bounds for approximate discrete tomography solutions

TL;DR

This paper extends algebraic discrete tomography theory to noisy measurements on arbitrary or convex finite sets, providing bounds on how closely binary solutions can approximate given line sums in multiple directions.

Contribution

It generalizes previous results to noisy and more general sets, and applies a method to bound the difference between approximate and exact line sums.

Findings

Existence of binary functions close to given line sums within bounds

Generalization to arbitrary and convex sets in b

Application of Beck and Fiala's method for approximation bounds

Abstract

In earlier papers we have developed an algebraic theory of discrete tomography. In those papers the structure of the functions $f: A \to \{0,1\}$ and $f: A \to \mathbb{Z}$ having given line sums in certain directions have been analyzed. Here $A$ was a block in $\mathbb{Z}^n$ with sides parallel to the axes. In the present paper we assume that there is noise in the measurements and (only) that $A$ is an arbitrary or convex finite set in $\mathbb{Z}^n$. We derive generalizations of earlier results. Furthermore we apply a method of Beck and Fiala to obtain results of he following type: if the line sums in $k$ directions of a function $h: A \to [0,1]$ are known, then there exists a function $f: A \to \{0,1\}$ such that its line sums differ by at most $k$ from the corresponding line sums of $h$.