Algorithms for linear time reconstruction by discrete tomography II

TL;DR

This paper introduces a linear time algorithm for reconstructing functions from line sums in discrete tomography, effective when function values are in a field or UFD, and addresses ambiguities with freely assigned values.

Contribution

It presents a novel linear time reconstruction algorithm that efficiently handles ambiguities in discrete tomography when function values are in a field or UFD.

Findings

Algorithm operates in linear time relative to directions and grid size.

Successfully reconstructs original function values outside switching domains.

Provides examples demonstrating effectiveness and ambiguity handling.

Abstract

The reconstruction of an unknown function $f$ from its line sums is the aim of discrete tomography. However, two main aspects prevent reconstruction from being an easy task. In general, many solutions are allowed due to the presence of the switching functions. Even when uniqueness conditions are available, results about the NP-hardness of reconstruction algorithms make their implementation inefficient when the values of $f$ are in certain sets. We show that this is not the case when $f$ takes values in a field or a unique factorization domain, such as $\R$ or $\Z$. We present a linear time reconstruction algorithm (in the number of directions and in the size of the grid), which outputs the original function values for all points outside of the switching domains. Freely chosen values are assigned to the other points, namely, those with ambiguities. Examples are provided.