Stability results for sets of uniqueness in binary tomography

Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano

^a Corresponding author: paolo.dulio@polimi.it
^b Corresponding author: silviamaria.pagani@polimi.it

Abstract

The recovery of an unknown density function from the knowledge of its projections is the aim of tomography. In many cases, considering the problem from a discrete perspective is more convenient than employing a continuous approach: discrete tomography, and in particular binary tomography, is therefore exploited. One of the main goals of tomography is guaranteeing that the produced output coincides with the scanned object, namely, one wants to achieve uniqueness of reconstruction, even when only a few directions, from which projections are taken, are employed. Relying on a theoretical result stating that special sets of just four lattice directions are enough to uniquely reconstruct a binary grid, we prove that such sets are stable, in the sense that a small discrete perturbation of the components of the directions returns sets which again ensure uniqueness of reconstruction. Examples are provided.