Uniqueness in Discrete Tomography of Planar Model Sets

TL;DR

This paper investigates the uniqueness of determining finite subsets of planar model sets, specifically cyclotomic model sets, using a limited number of directional X-ray measurements, with implications for discrete tomography.

Contribution

It establishes that convex subsets of cyclotomic model sets are uniquely determined by four specific X-ray directions, and any finite subset can be successively determined by just two directions.

Findings

Convex subsets are determined by four X-ray directions.

Any finite subset can be successively determined by two directions.

Results are illustrated with examples like square and triangle tilings.

Abstract

The problem of determining finite subsets of characteristic planar model sets (mathematical quasicrystals) $\varLambda$, called cyclotomic model sets, by parallel $X$-rays is considered. Here, an $X$-ray in direction $u$ of a finite subset of the plane gives the number of points in the set on each line parallel to $u$. For practical reasons, only $X$-rays in $\varLambda$-directions, i.e., directions parallel to non-zero elements of the difference set $\varLambda - \varLambda$, are permitted. In particular, by combining methods from algebraic number theory and convexity, it is shown that the convex subsets of a cyclotomic model set $\varLambda$, i.e., finite sets $C\subset \varLambda$ whose convex hulls contain no new points of $\varLambda$, are determined, among all convex subsets of $\varLambda$, by their $X$-rays in four prescribed $\varLambda$-directions, whereas any set of three $\varLambda$-directions does not suffice for this purpose. We also study the interactive technique of successive determination in the case of cyclotomic model sets, in which the information from previous $X$-rays is used in deciding the direction for the next $X$-ray. In particular, it is shown that the finite subsets of any cyclotomic model set $\varLambda$ can be successively determined by two $\varLambda$-directions. All results are illustrated by means of well-known examples, i.e., the cyclotomic model sets associated with the square tiling, the triangle tiling, the tiling of Ammann-Beenker, the T\"ubingen triangle tiling and the shield tiling.