Uniqueness in Discrete Tomography: Three Remarks and a Corollary

Abstract

Discrete tomography is concerned with the retrieval of finite point sets in some $\mathbbm{R}^d$ from their X-rays in a given number m of directions $u_1,\ldots, u_m$. In the present paper we focus on uniqueness issues. The first remark gives a uniform treatment and extension of known uniqueness results. In particular, we introduce the concept of J-additivity and give conditions when a subset J of possible positions is already determined by the given data. As a by-product, we settle a conjecture of Brunetti and Daurat on planar lattice convex sets. Remark 2 resolves a problem of Kuba posed in 1997 on the uniqueness in the case $d=m=3$ with $u_1,u_2,u_3$ being the standard unit vectors. Remark 3 determines the computational complexity of finding a smallest set J of positions whose disclosure yields uniqueness. As a corollary, we obtain a hardness result for 0-1-polytopes.