It is a classic problem to estimate continuous-time sparse signals, like point sources in a direction-of-arrival problem, or pulses in a time-of-flight measurement. The earliest occurrence is the estimation of sinusoids in time series using Prony's method. This is at the root of a substantial line of work on high resolution spectral estimation. The estimation of continuous-time sparse signals from discrete-time samples is the goal of the sampling theory for finite rate of innovation (FRI) signals. Both spectral estimation and FRI sampling usually assume uniform sampling. But not all measurements are obtained uniformly, as exemplified by a concrete radioastronomy problem we set out to solve. Thus, we develop the theory and algorithm to reconstruct sparse signals, typically sum of sinusoids, from  nonuniform  samples. We achieve this by identifying a linear transformation that relates the unknown uniform samples of sinusoids to the given measurements. These uniform samples are known to satisfy the annihilation equations. A valid solution is then obtained by solving a constrained minimization such that the reconstructed signal is consistent with the given measurements and satisfies the annihilation constraint. Thanks to this new approach, we unify a variety of FRI-based methods. We demonstrate the versatility and robustness of the proposed approach with five FRI reconstruction problems, namely Dirac reconstructions with irregular time or Fourier domain samples, FRI curve reconstructions, Dirac reconstructions on the sphere, and point source reconstructions in radioastronomy. The proposed algorithm improves substantially over state-of-the-art methods and is able to reconstruct point sources accurately from irregularly sampled Fourier measurements under severe noise conditions.
['Hanjie Pan', 'Thierry Blu', 'Martin Vetterli']
Towards Generalized FRI Sampling With an Application to Source Resolution in Radioastronomy