Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum

Abstract

The purpose of this paper is to report on recent approaches to reconstruction problems based on analog, or in other words, infinite-dimensional, image and signal models. We describe three main contributions to this problem. First, linear reconstructions from sampled measurements via so-called generalized sampling (GS). Second, the extension of generalized sampling to inverse and ill-posed problems. And third, the combination of generalized sampling with sparse recovery techniques. This final contribution leads to a theory and set of methods for infinite-dimensional compressed sensing, or as we shall also refer to it, compressed sensing over the continuum.