A multi-parameter optimization approach for complex continuous sparse modelling

The main focus of this work is the estimation of a complex valued signal assumed to have a sparse representation in an uncountable dictionary of signals. The dictionary elements are parameterized by a real-valued vector and the available observations are corrupted with an additive noise. By applying a linearization technique, the original model is recast as a constrained sparse perturbed model. The problem of the computation of the involved multiple parameters is addressed from a nonconvex optimization viewpoint. A cost function is defined including an arbitrary Lipschitz differentiable data fidelity term accounting for the noise statistics, and an ℓ₀-like penalty. A proximal algorithm is then employed to solve the resulting nonconvex and nonsmooth minimization problem. Experimental results illustrate the good practical performance of the proposed approach when applied to 2D spectrum analysis.