An Algebraic Framework for Discrete Tomography: Revealing the Structure of Dependencies

Discrete tomography is concerned with the reconstruction of images that are defined on a discrete set of lattice points from their projections in several directions. The range of values that can be assigned to each lattice point is typically a small discrete set. In this paper we present a framework for studying these problems from an algebraic perspective, based on Ring Theory and Commutative Algebra. A principal advantage of this abstract setting is that a vast body of existing theory becomes accessible for solving Discrete Tomography problems. We provide proofs of several new results on the structure of dependencies between projections, including a discrete analogon of the well-known Helgason-Ludwig consistency conditions from continuous tomography.