Sampling, sparsity, and inverse problems

Sampling is a central topic in signal processing, communications, and in all fields where the world is analog and computation is digital. The question is simple: When does a countable set of measurements allow a perfect and stable representation of a class of signals? This allows the reconstruction of the analog world, or interpolation. A related problem is when these measurements allow to solve inverse problems accurately, like source localization. Classic results concern bandlimited functions and shift-invariant subspaces, and use linear approximation. Recently, nonlinear methods have appeared, based on parametric methods and/or convex relaxation, which allow a broader class of sampling results. We review sampling of finite rate of innovation (FRI) signals, which are non-bandlimited continuous-time signals with a finite parametric representation. This leads to sharp results on sampling and reconstruction of such sparse continuous-time signals. We then explore performance bounds on retrieving sparse continuous-time signals buried in noise. While this is a classic estimation problem, we show sharper lower bounds for simple cases, indicating (i) there is a phase transition and (ii) current algorithms are close to the bounds. This leads to notions of resolution or resolvability. We then turn our attention to sampling problems where physics plays a central role. After all, many sensed signals are the solution of some PDE. In these cases, continuous-time or continuous-space modeling can be advantageous, be it to reduce the number of sensors and/or the sampling rate. First, we consider the wave equation, and review the fact that wave fields are essentially bandlimited in space-time domain. This can be used for critical sampling of acquisition or rendering of wave fields. We also show an acoustic source localization problem, where wideband frequency probing and finite element modeling show interesting localization power. Then, in a diffusion equation scenario, source loc- lization using a sensor network can be addressed with a parametric approach, indicating trade-offs between spatial and temporal sampling densities. This can be used in air pollution monitoring and temperature sensing. In all these problems, the computational tools like FRI or CS come in handy when the modeling and the conditioning is adequate. Last but not least, the proof of the pudding is in experiments and/or real data sets.