Compressive Imaging: from advanced signal processing theory to astronomy and medicine

Abstract: In this talk we will discuss some of the latest evolutions in compressive imaging. We will start with a review of the basics of the theory of compressive sampling, which demonstrates that sparse signals may be recovered accurately and robustly from a number of random and incoherent measurements scaling with the sparsity rather than the ambient dimension. We will pursue by highlighting two novel compressive sampling approaches, one at the level of signal acquisition, the other at the level signal reconstruction. The acquisition method promoted, coined spread spectrum, consists in pre-modulating signals by random phases prior to random Fourier under-sampling. The reconstruction approach highlighted acknowledges the fact that signals are often sparse in multiple frames, and promotes an average sparsity prior to regularise the ill-posed inverse problem for signal recovery. It is dubbed Sparsity Averaging Reweighted Analysis (SARA) and relies on nonlinear and iterative convex optimisation algorithms. We will finally discuss the application of these techniques and illustrate their drastic superiority relative to the state of art in two fields of major importance in astronomy and medicine, respectively interferometric imaging and magnetic resonance imaging.