Revisiting sparse ICA from a synthesis point of view: Blind Source Separation for over and underdetermined mixtures

This paper studies the existing links between two approaches of Independent Component Analysis (ICA), projection pursuit and Infomax/maximum likelihood estimation, and the Sparse Component Analysis (SCA), mainly used in the Generalized Morphological Component Analysis (GMCA), to tackle the Blind Source Separation (BSS) of instantaneous mixtures problem. If ICA methods suit well for overdetermined and noiseless mixtures, SCA (via GMCA) has demonstrated its robustness to noise. Using the "synthesis" point of view to reformulate ICA methods as an optimization problem, we propose a new optimization framework, which encompasses both approaches. We show that the algorithms developed to minimize the proposed functional built on SCA, but imposing a numerical decorrelation constraint on the sources, aims to improve the Signal to Inference Ratio (SIR) of the estimated sources, without degrading the Signal to Distortion Ratio (SDR).