On the computational complexity of reconstructing lattice sets from their X-rays

We study the computational complexity of various inverse problems in discrete tomography. These questions are motivated by demands from the material sciences for the reconstruction of crystalline structures from images produced by quantitative high resolution transmission electron microscopy. We completely settle the complexity status of the basic problems of existence (data consistency), uniqueness (determination), and reconstruction of finite subsets of the d-dimensional integer lattice ℤ^d that are only accessible via their line sums (discrete X-rays) in some prescribed finite set of lattice directions. Roughly speaking, it turns out that for all d ≥ 2 and for a prescribed but arbitrary set of m ≥ 2 pairwise nonparallel lattice directions, the problems are solvable in polynomial time if m = 2 and are ℕℙ-complete (or ℕℙ-equivalent) otherwise.