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

Andreea Denitiu, Stefania Petra, Claudius Schnörr, Christoph Schnoerr

Abstract:

We study unique recovery of cosparse signals from limited-view tomographic measurements of two- and three-dimensional domains. Admissible signals belong to the union of subspaces defined by all cosupports of maximal cardinality $l$ with respect to the discrete gradient operator.
We relate $l$ 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.

Keywords: compressed sensing, underdetermined systems of linear equations, cosparsity, total vari-ation, discrete and limited-view tomography.