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

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 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.

Testable uniqueness conditions for empirical assessment of undersampling levels in total variation-regularized X-ray CT

“…It is generally acknowledged, however, that existing guarantees either do not apply to or give extremely pessimistic bounds in deterministic sampling contexts [10]. In particular for CT, [11,12,13] describe the lack of general guarantees, while [11,12] derive preliminary average-case results for certain restricted special geometries known as discrete geometry; however these results do not cover regular sampling patterns in CT, such as parallel-beam and fan-beam geometries.…”

An Entropic Perturbation Approach to TV-Minimization for Limited-Data Tomography

“…The above lower bounds on the number of measurements m required to recover a -cosparse vector u, imply that recovery can be carried out by solving min u Bu 0 subject to Au = b. Replacing this objective by the convex relaxation (2) yields an excellent agreement of empirical results with the prediction (5), as shown in [2], although the independency assumption made in Prop. (1) does not strictly hold for the sensor matrices A and the operator B (2) used in practice.…”

“…We summarize in this section an essential result from [2] concerning the unique recovery from tomographic projection by solving problem (1), depending on the cosparsity level of u (Definition 1) and the number m of measurements (projections) of u. This result motivates the mathematical programming approach discussed in Section 3 and the corresponding numerical optimization approach presented in Section 4.…”

“…Thus, B Λ u = 0. In view of the specific operator B given by (2), measures the homogeneity of the volume function u. A large value of can be expected for u corresponding to solid bodies composed of few homogeneous components.…”

“…By adopting the cosparse signal model [1] that conforms to our tomographic scenario, the authors [2] recently established the existence of "phase transitions" that relate the required limited(!) number of measurements to the cosparsity level of u in order to guarantee unique recovery.…”