A linear time approach to three-dimensional reconstruction by discrete tomography

TL;DR

This paper presents a linear-time method for three-dimensional discrete tomography reconstruction under nonproportionality conditions, extending previous 2D results to 3D boundary ghosts, enabling efficient solution characterization.

Contribution

It introduces a linear-time approach for 3D reconstruction in discrete tomography, generalizing 2D methods to handle boundary ghosts.

Findings

Linear-time solution exists for 3D case under nonproportionality.

Method effectively characterizes all solutions, including ghosts.

Applicable to boundary ghost scenarios in 3D reconstruction.

Abstract

The goal of discrete tomography is to reconstruct an unknown function $f$ via a given set of line sums. In addition to requiring accurate reconstructions, it is favourable to be able to perform the task in a timely manner. This is complicated by the presence of ghosts, which allow many solutions to exist in general. In this paper we consider the case of a function $f : A \to \mathbb{R}$ where $A$ is a finite grid in $\mathbb{Z}^3$. Previous work has shown that in the two-dimensional case it is possible to determine all solutions in parameterized form in linear time (with respect to the number of directions and the grid size) regardless of whether the solution is unique. In this work, we show that a similar linear method exists in three dimensions under the condition of nonproportionality. We show that the condition of nonproportionality is fulfilled in the case of three-dimensional boundary ghosts.