Phase Transitions and Cosparse Tomographic Recovery of Compound Solid Bodies from Few Projections

TL;DR

This paper investigates the conditions under which cosparse signals can be uniquely recovered from limited-angle tomographic measurements, providing theoretical insights and numerical validation relevant to practical non-invasive inspection of solid objects.

Contribution

It establishes new nullspace and measurement conditions for unique cosparse signal recovery in limited-angle tomography, supported by comprehensive numerical experiments.

Findings

High correlation between theoretical predictions and practical performance

Unique recovery is feasible despite poor measurement matrix properties

Applicable to contactless quality inspection of compound solids

Abstract

We study unique recovery of cosparse signals from limited-angle tomographic measurements of two- and three-dimensional domains. Admissible signals belong to the union of subspaces defined by all cosupports of maximal cardinality $\ell$ with respect to the discrete gradient operator. We relate $\ell$ both to the number of measurements and to a nullspace condition with respect to the measurement matrix, so as to achieve unique recovery by linear programming. These results are supported by comprehensive numerical experiments that show a high correlation of performance in practice and theoretical predictions. Despite poor properties of the measurement matrix from the viewpoint of compressed sensing, the class of uniquely recoverable signals basically seems large enough to cover practical applications, like contactless quality inspection of compound solid bodies composed of few materials.