Sparse Component Analysis: A General Framework for Linear and Nonlinear Blind Source Separation and Mixture Identification

Abstract

In this chapter, we consider two closely related data processing tasks. The first one is Blind Source Separation (BSS), which consists in estimating a set of unknown source data (one-dimensional signals, images, ...) from observed mixtures of these data, while the mixing operator has unknown parameter values. The second task is Blind Mixture Identification (BMI), which aims at estimating these unknown parameter values of the mixing operator. We provide a unified view and describe the latest extensions of the general framework that we have been developing for BSS and BMI since the beginning of the 2000s. This framework yields a wide range of BSS/BMI methods applicable to various types of sources (one-dimensional signals, images, ...) mixed according to various models (linear instantaneous, anechoic, full convolutive, nonlinear and especially linear-quadratic), possibly with non-negativity or sum-to-one constraints. This framework is based on the concept of joint sparsity of the source data, considered in various domains (original temporal or spatial domain, transformed representation in time-frequency or time-scale/wavelet domain, ...). More precisely, the proposed methods essentially require a few tiny zones, in mixed signals or in their transformed versions, where only one of the source “signals” is active, i.e., nonzero. They therefore set very limited constraints on source sparsity and could then be considered as “quasi-non-sparse component analysis” methods. Besides, unlike Independent Component Analysis methods, they are suited to correlated sources. We also discuss their application to various data processing functions, ranging from audio signal separation to unmixing of hyperspectral remote sensing images.