Sparse BSS from Poisson Measurements

The problem of sparse Blind Source Separation (BSS) has been extensively studied when the noise is additive and Gaussian. This is however not the case when the measurements follow Poisson or shot noise statistics, which is customary with counting-based measurements. To that purpose, we introduce a novel sparse BSS algorithm coined pGMCA (poisson-Generalized Morphological Component Analysis) that specifically tackles the blind separation of sparse sources from measurements following Poisson statistics. The proposed algorithm builds upon Nesterov's smoothing technique to define a smooth approximation of sparse BSS, with a data fidelity term derived from the Poisson likelihood. This allows to design a block coordinate descent-based minimization procedure with a simple choice of the regularization parameter. Numerical experiments have been carried out that illustrate the robustness of the proposed method with respect to Poisson noise. The pGMCA algorithm has been further evaluated in a realistic astrophysical X-ray imaging setting.