An Entropic Perturbation Approach to TV-Minimization for Limited-Data Tomography
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 ℓ1-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 ℓ1-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.
Keywordsdiscrete tomography compressed sensing underdetermined systems of linear equations cosparsity phase transitions total variation entropic perturbation convex duality convex programming
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