Data structures for robust multifrequency imaging

In this paper, we consider imaging problems that can be cast in the form of an underdetermined linear system of equations. When a single measurement vector is available, a sparsity promoting ℓ1-minimization-based algorithm may be used to solve the imaging problem efficiently. A suitable algorithm in the case of multiple measurement vectors would be the MUltiple SIgnal Classification (MUSIC) which is a subspace projection method. We provide in this work a theoretical framework in an abstract linear algebra setting that allows us to examine under what conditions the ℓ1-minimization problem and the MUSIC method admit an exact solution. We also examine the performance of these two approaches when the data are noisy. Several imaging configurations that fall under the assumptions of the theory are discussed such as active imaging with single- or multiple-frequency data. We also show that the phase-retrieval problem can be re-cast under the same linear system formalism using the polarization identity and relying on diversity of illuminations. The relevance of our theoretical analysis in imaging is illustrated with numerical simulations and robustness to noise is examined by allowing the background medium to be weakly inhomogeneous.