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

TL;DR

This paper introduces an algebraic framework for discrete tomography using ring and commutative algebra, enabling new insights into the structure of dependencies between projections and extending classical consistency conditions.

Contribution

It develops a novel algebraic approach to discrete tomography, connecting it with established algebraic theories and deriving new dependency results.

Findings

Established an algebraic structure for discrete tomography problems.

Proved a discrete analog of Helgason-Ludwig consistency conditions.

Enhanced understanding of projection dependencies in discrete images.

Abstract

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.