Uniqueness in Discrete Tomography of Delone Sets with Long-Range Order

TL;DR

This paper investigates how to uniquely determine finite subsets of Delone sets with long-range order using X-ray measurements in specific directions, introducing algebraic Delone sets and conditions for convex subset identification.

Contribution

It introduces the concept of algebraic Delone sets and provides a sufficient condition for uniquely determining convex subsets via four specific X-ray directions.

Findings

Defined algebraic Delone sets in 2

Established a sufficient condition for convex subset determination

Achieved uniqueness results with four 2-directions

Abstract

We address the problem of determining finite subsets of Delone sets $\varLambda\subset\R^d$ with long-range order by $X$-rays in prescribed $\varLambda$-directions, i.e., directions parallel to non-zero interpoint vectors of $\varLambda$. Here, an $X$-ray in direction $u$ of a finite set gives the number of points in the set on each line parallel to $u$. For our main result, we introduce the notion of algebraic Delone sets $\varLambda\subset\R^2$ and derive a sufficient condition for the determination of the convex subsets of these sets by $X$-rays in four prescribed $\varLambda$-directions.