Cadzow Denoising Upgraded: A New Projection Method for the Recovery of Dirac Pulses from Noisy Linear Measurements

We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpass-filtered samples. We show that maximum-likelihood estimation of the unknown parameters amounts to a difficult, even believed NP-hard, matrix problem of structured low rank approximation. To solve it, we propose a new heuristic iterative algorithm, based on a recently proposed splitting method for convex nonsmooth optimization. Although the algorithm comes, in absence of convexity, with no convergence proof, it converges in practice to a local solution, and even to the global solution of the problem, when the noise level is not too high. Thus, our method improves upon the classical Cadzow denoising method, for same ease of implementation and speed.