CPGD: Cadzow Plug-and-Play Gradient Descent for Generalised FRI

Finite rate of innovation (FRI) is a powerful reconstruction framework enabling the recovery sparse Dirac streams from uniform low-pass filtered samples. An extension this framework, called generalised FRI (genFRI), has been recently proposed for handling cases with arbitrary linear measurement models. In context, signal amounts to solving joint constrained optimisation problem, yielding estimates both Fourier series coefficients stream and its so-called annihilating filter, involved in regularisation term. This problem however highly non convex data. Moreover, numerical solver computationally intensive without convergence guarantee. work, we propose an implicit formulation genFRI problem. To end, leverage novel term which does not depend explicitly on unknown filter yet enforces sufficient structure solution stable recovery. The resulting still convex, but simpler since data less unknowns. We solve it by means provably convergent proximal gradient descent (PGD) method. Since step admit simple closed-form expression, inexact PGD method, coined as Cadzow plug-and-play (CPGD). latter approximates steps denoising, well-known denoising algorithm FRI. provide local fixed-point guarantees CPGD. Through extensive simulations, demonstrate superiority CPGD against state-of-the-art case time