Bounds for discrete tomography solutions

TL;DR

This paper studies the mathematical bounds and stability of solutions in discrete tomography, providing explicit formulas and estimates for the distance between solutions, with applications to real, integer, and torus cases.

Contribution

It introduces explicit projection formulas, distance estimates, and stability bounds for solutions in discrete tomography, including generalizations to torus and continuous cases.

Findings

Explicit projection vector formula for row and column sums.

Method to estimate maximum distance between binary solutions.

Upper bounds for distance from real to integer solutions.

Abstract

We consider the reconstruction of a function on a finite subset of $\mathbb{Z}^2$ if the line sums in certain directions are prescribed. The real solutions form a linear manifold, its integer solutions a grid. First we provide an explicit expression for the projection vector from the origin onto the linear solution manifold in the case of only row and column sums of a finite subset of $\mathbf{Z}^2$. Next we present a method to estimate the maximal distance between two binary solutions. Subsequently we deduce an upper bound for the distance from any given real solution to the nearest integer solution. This enables us to estimate the stability of solutions. Finally we generalize the first mentioned result to the torus case and to the continuous case.