Uniqueness in discrete tomography: Three remarks and a corollary

Discrete tomography is concerned with the retrieval of finite point sets in some d from their X-rays in a given number m of directions u1, . . . , um. 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 u1, u2, u3 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.