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

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