Blind Separation of Noisy Multivariate Data Using Second-Order Statistics: Remote-Sensing Applications

Abstract

In this paper a second-order method for blind source separation of noisy instantaneous linear mixtures is presented for the case where the signal order <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> is unknown. Its performance advantages are illustrated by simulations and by application to Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) multichannel visible/infrared data. The model assumes that <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> mixtures <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</b> of dimension <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> are observed, where <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</b> = <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ap</b> + <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Gw</b> , and the underlying signal vector <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</b> has k < n/3 independent unit-variance elements. <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> is the mixing matrix, <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</b> is diagonal, and <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</b> is a normalized white-noise vector. The algorithm estimates the Second-Order separation matrix <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> , signal Order <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> , and Noise and is therefore designated as SOON. SOON first iteratively estimates <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> and <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</b> using a scree metric, singular-value decomposition, and the expectation-maximization algorithm, and then determines the values of <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AP</b> and <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</b> . The final step estimates <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> and the set of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> signal vectors <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</b> using a variant of the joint-diagonalization method used in the Second-Order Blind Identification (SOBI) and Second-Order NonStationary (SONS) source-separation algorithms. The SOON extension of SOBI and SONS significantly improves their separation of simulated sources hidden in noise. SOON also reveals interesting thermal dynamics within AVIRIS multichannel visible/infrared imaging data not found by noise-adjusted principal-component analysis.