Compressive Sensing and Signal Subspace Methods for Inverse Scattering including Multiple Scattering

Compressive sensing is a new field in signal processing and applied mathematics. It allows one to simultaneously sample and compress signals which are known to have a sparse representation in a known basis or dictionary along with the subsequent recovery by linear programming (requiring polynomial (P) time) of the original signals with low or no error [1, 2, 3]. In a discrete setting, sparsity means that a vector, say V (L× 1), can be expanded as V = Bτ where the L×L matrix B represents the basis or dictionary and the L× 1 vector τ is sparse, i.e., most of its entries (the expansion coefficients) are zero (negligible in practice). Compressive measurements or samples are non-adaptive, possibly random linear projections of the given signal. Most importantly, sparsity arises in many physical signals, hence this approach is of significant importance. The results in this area apply to magnetic resonance imaging, astronomy, single-pixel photography, and other disciplines. The present research develops new Bayesian-compressive-sensing-based algorithms for the inverse scattering problem within exact scattering theory (where the forward mapping from object function to scattering matrix is nonlinear). In these methods, a 1-norm regularizing constraint substitutes the 2-norm constraint that is more typically adopted in inverse theory. This 1-norm constraint is typical of compressive sensing theory [1, 2, 3]. It is an approach to incorporate a priori knowledge on sparsity of the sought-after solution. This work also reports new signal-subspace-based imaging algorithms for the shape reconstruction or inverse support aspect of the full inverse scattering problem of estimating the object function from the scattering matrix. In the present communication we outline the main ideas behind the new Bayesian compressive sensing approach to inverse scattering but the talk at the conference will comparatively address both the compressive sensing and the signal-subspace-based approaches. We outline next the key steps to derive new inverse scattering approaches based on maximum a posteriori probability (MAP) estimators for the unknown object function which is assumed to arise as a realization of a sparsity-inducing Laplace prior. They are compressive sensing counterparts of conventional iterative Born methods. Other Bayesian compressive sensing and signal subspace methods for imaging will be presented at the talk, along with numerical illustrations.