Reconstructing Polyatomic Structures from Discrete X-Rays: NP-Completeness Proof for Three Atoms

Marek Chrobak, Christoph Dürr

Abstract: We address a discrete tomography problem that arises in the study of the atomic structure of crystal lattices. A polyatomic structure T can be defined as an integer lattice in dimension D≥2, whose points may be occupied by c distinct types of atoms. To “analyze” T, we conduct ℓ measurements that we call discrete X-rays. A discrete X-ray in direction ξ determines the number of atoms of each type on each line parallel to ξ. Given ℓ such non-parallel X-rays, we wish to reconstruct T.