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 with respect to the discrete gradient operator. We relate 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.

The reconstruction problem of discrete tomography is studied using novel techniques from compressive sensing. Recent theoretical results of the authors enable to predict the number of measurements required for the unique reconstruction of a class of cosparse dense 2D and 3D signals in severely undersampled scenarios by convex programming. These results extend established 1-related theory based on sparsity of the signal itself to novel scenarios not covered so far, including tomographic projections of 3D solid bodies composed of few different materials. As a consequence, the large-scale optimization task based on total-variation minimization subject to tomographic projection constraints is considerably more complex than basic 1-programming for sparse regularization. We propose an entropic perturbation of the objective that enables to apply efficient methodologies from unconstrained optimization to the perturbed dual program. Numerical results validate the theory for large-scale recovery problems of integer-valued functions that exceed the capacity of the commercial MOSEK software.

No abstract