A geometrical blind separation method for unconstrained-sum locally dominant sources

This paper presents a geometrical method for solving the non-negative Blind Source Separation (BSS) problem. The method is based on a weak sparsity condition: for each source, there should exist at least one observed vector where only this source is non-zero. The method does not require but allows the sum-to-one constraint for the mixing parameters or sources. Considering each observed vector as an element of a vector space, the scatter plot of these vectors yields a simplicial cone. The identification of the edges of this simplicial cone provides an estimate of the scaled mixing coefficients. The observations being non-negative, our approach to find this simplical cone is to identify the observed vectors which are furthest apart in the angular sense. The sources are finally reconstructed from the observed vectors and mixing coefficients with a non-negative least square algorithm. The proposed method called MASS (Maximum Angle Source Separation) is effective for possibly correlated but linearly independent sources. Various tests on synthetic data show the good performance of this approach and an experiment on real hyperspectral data illustrates its efficiency.