Computational complexity of three-dimensional discrete tomography with missing data

Abstract

Discrete tomography deals with problems of determining shape of a discrete object from a set of projections. In this paper, we deal with a fundamental problem in discreet tomography: reconstructing a discrete object in $$\mathbb {R}^3$$ from its orthogonal projections, which we call three-dimensional discrete tomography. This problem has been mostly studied under the assumption that complete data of the projections are available. However, in practice, there might be missing data in the projections, which come from, e.g., the lack of precision in the measurements. In this paper, we consider the three-dimensional discrete tomography with missing data. Specifically, we consider the following three fundamental problems in discrete tomography: the consistency, counting, and uniqueness problems, and classify the computational complexities of these problems in terms of the length of one dimension. We also generalize these results to higher-dimensional discrete tomography, which has applications in operations research and statistics.