Subspace and Bayesian Compressive Sensing Methods in Imaging

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]. 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 biomedical imaging, astronomy, single-pixel photography [4, 5], and many other disciplines. The present work presents new approaches to linear and nonlinear inverse scattering problems that are based on compressive sensing ideas. Particular emphasis is given to the framework termed Bayesian compressive sensing [1]. Past work in compressive sensing has been restricted to linear inverse problems of the form y = Ax where A is a matrix mapping input (object) x to output (data) y. In this linear context, the focus has been to show that despite significant undersampling of the data signal y as projections of the form y0 = Φy where Φ is a measurement matrix obeying a mild incoherency property, one can for a broad class of sparse object signals still carry out perfect inversions or reconstructions with low error to the compressed inverse problem of inverting x from y0 where y0 = Φy = ΦAx (refer to [2, 3] for the details). The focus of the present treatment is to show how Bayesian compressive sensing applies to wave inverse scattering, in both the linear regime of the so-called Born approximation for weakly scattering objects as well as in the more general context of strongly scattering objects exhibiting non-negligible multiple scattering interactions. The compressive sensing approach to inverse scattering will be discussed in relation to alternative signal-subspace-based methods for shape reconstruction including sampling and level-set-based approaches [6, 7, 8]. The derived developments are motivated mostly by detection and imaging applications requiring computationally non-intensive (P time) signal processing with limited wave data about a given event or target of interest.