The problem of reconstructing or estimating partially observed orsampled signals is animportant one that finds applicationin many areas of signal processing and communications. Traditionalacquisition and reconstruction approaches are heavily influences byclassical Shannon sampling theory which gives an exact samplingand interpolation formula for bandlimited signals. Recently, theemerging theory of sparse sampling has challenged the waywe think about signal acquisition and has demonstrated that, byusing more sophisticated signal models, it is possible to break awayfrom the need to sample signals at the Nyquist rate.The insight thatsub-Nyquist sampling can, under some circumstances, allow perfectreconstruction is revolutionizing signal processing, communicationsand inverse problems.Given theubiquity of the sampling process, the implications of these newresearch developments are far reaching.This project is based on the applicant's recent work on the samplingof sparse continuous-time signals and aims to extend the existing theory to include moregeneral signal models that are closer to the physicalcharacteristics of real data, to explore new domains where sparsityand sampling can be effectively used and to provide a setof new fast algorithms with clear and predictable performance.Aspart of this work, he will also consider timely important problemssuch as the localization of diffusive sources in sensor networks andthe analysis of neuronal signals of the brain. He will, for thefirst time, pose these as sparse sampling problems and in this wayhe expects to develop technologies with a step change inperformance.