A geometric approach to blind separation of nonnegative and dependent source signals

Abstract

Blind source separation (BSS) consists in processing a set of observed mixed signals to separate them into a set of original components. Most of the current blind separation methods assumes that the source signals are "as statistically independent as possible". In many real-world cases, however, source signals are considerably dependent. In order to cope with such signals, we proposed in [1] a geometric method that separates dependent signals provided that they are nonnegative and locally orthogonal. This paper also presents a geometric method for separating nonnegative source signals which relies on an assumption weaker than local orthogonality. The separation problem relies on the identification of relevant facets of the data cone. After a rigorous proof of the proposed method, we give the details of the separation algorithm and report experiments carried out on signals from various origins, clearly showing the contribution of our method.

A Convex Geometry-Based Blind Source Separation Method for Separating Nonnegative Sources

“…In this section, we provide simulation examples to illustrate the performance of the proposed CG-based algorithm, in comparison with the NICA algorithm [7], the DIEM algorithm [12], the DEDS algorithm [35], the CAMNS-LP algorithm [36], and the VCA algorithm [37]. With [36], the source separation performance is measured by the sum square error (M-SSE) index defined as follows:…”

“…However, these methods have local minima problem that results from using the alternative least-square iteration optimization scheme, and thus, BSS is not a guarantee. Similarly, by using the nonnegativity of sources, the method in [35] does not depend on statistical features of the sources. Specifically, it is shown [35] that one can obtain a finite set of candidate source signals that contain the original source signals, and the latter can be identified if they are the most linearly independent (MLI) among the respective set of candidate source signals.…”

Multiscale convolutive blind source separation in wavelet transform domain

“…The main contribution of this paper is the use of an enhanced wavelet packet transforms instead of STFT and FFT, together with a well suited geometric unmixing algorithm [8]. This work is motivated by the one presented in [9], where it was demonstrated that the focus on the sparseness of the sources and their mixtures, once they are projected onto a proper space of sparse representation.…”

“…the model described by Equation (1), we briefly describe a geometric algorithm dedicated to separating nonnegative source signals that may be highly correlated [8]. In addition to being nonnegative, the source signals must also verify a second hypothesis which can be stated as follows Hypothesis 1.…”

“…This step usually results in a rectangular matrix, which will be denoted by X * . The main theoretical contribution of the [8] is a theorem which states that thep true source signals are among the columns of the matrix product given by X T X * . Hence, the columns of X T X * provide a finite set of candidate source signals.…”