Sparsity and Coherence in Inverse Problems: from 1 D to 2D Processing through Matrix Completion

Sparseness-promoting inversion strategies based on Bayesian formulations have recently emerged as powerful, robust, flexible, and accurate techniques to address a wide variety of problems arising in microwave imaging. The success of such paradigm is motivated by the capability of the resulting formulations to (i) handle sparse linear/linearized problems even when incoherence or restricted isometry properties (RIPs) are not guaranteed, and to (ii) provide self-computed “confidence level” estimations. Unfortunately, no methodology has been proposed in the state-of-the-art to actually exploit such confidence-level indexes and enhance the reconstructed profiles once the imaging reliability is self-detected to be poor (e.g., in harsh signal-to-noise-ratio scenarios, or when the RIP does not hold true). Moreover, the most popular BCS formulations handle the unknowns as 1D vectors, therefore neglecting the “matrix” nature of the associated physical problem (e.g., contrast retrieval). In this framework, this paper is aimed at illustrating the possibility to exploit a new class of methodologies based on rank-minimization formulations to overcome the above limitations and recover reliable reconstructions of the scattering scenario by fully exploiting the 2D matrix nature of the problem. Selected preliminary numerical results will be also discussed to validate the proposed paradigm.