On the difference between solutions of discrete tomography problems II

TL;DR

This paper investigates the differences between solutions to binary image reconstruction from projections, establishing bounds on their symmetric difference and demonstrating the optimality of these bounds.

Contribution

It provides a lower bound on the symmetric difference between solutions and proves that this bound is tight, extending previous work on solution uniqueness.

Findings

Lower bound of 2A+2 on symmetric difference between solutions

Constructive proof showing the bound is tight

Extension of previous upper bound results

Abstract

We consider the problem of reconstructing binary images from their horizontal and vertical projections. It is known that the projections do not necessarily determine the image uniquely. In a previous paper it was shown that the symmetric difference between two solutions (binary images that satisfy the projections) is at most 4A times the square root of 2N. Here N is the sum of the projections in one direction (i.e. the size of the image) and A is a parameter depending on the projections. In this paper we give a lower bound: for each set of projections that has at least two solutions, we construct two solutions that have a symmetric difference of at least 2A+2. We also show that this is the best possible.