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Blind Separation of QuasiStationary Sources: Exploiting Convex Geometry in Covariance Domain
, 2015
"... This paper revisits blind source separation of instantaneously mixed quasistationary sources (BSSQSS), motivated by the observation that in certain applications (e.g., speech) there exist time frames during which only one source is active, or locally dominant. Combined with nonnegativity of sourc ..."
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This paper revisits blind source separation of instantaneously mixed quasistationary sources (BSSQSS), motivated by the observation that in certain applications (e.g., speech) there exist time frames during which only one source is active, or locally dominant. Combined with nonnegativity of source powers, this endows the problem with a nice convex geometry that enables elegant and efficient BSS solutions. Local dominance is tantamount to the socalled pure pixel/separability assumption in hyperspectral unmixing/nonnegative matrix factorization, respectively. Building on this link, a very simple algorithm called successive projection algorithm (SPA) is considered for estimating the mixing system in closed form. To complement SPA in the specific BSSQSS context, an algebraic preprocessing procedure is proposed to suppress shortterm source crosscorrelation interference. The proposed procedure is simple, effective, and supported by theoretical analysis. Solutions based on volume minimization (VolMin) are also considered. By theoretical analysis, it is shown that VolMin guarantees perfect mixing system identifiability under an assumption more relaxed than (exact) local dominance—which means wider applicability in practice. Exploiting the specific structure of BSSQSS, a fast VolMin algorithm is proposed for the overdetermined case. Careful simulations using real speech sources showcase the simplicity, efficiency, and accuracy of the proposed algorithms.
WHEN CAN THE MINIMUM VOLUME ENCLOSING SIMPLEX IDENTIFY THE ENDMEMBERS CORRECTLY WHEN THERE IS NO PURE PIXEL?
"... In blind hyperspectral unmixing, it has been commonly believed that the minimum volume enclosing simplex (MVES) criterion is robust against lack of pure pixels. Specifically, such a belief has been based on empirical experience, where extensive numerical results showed that MVESbased algorithms may ..."
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In blind hyperspectral unmixing, it has been commonly believed that the minimum volume enclosing simplex (MVES) criterion is robust against lack of pure pixels. Specifically, such a belief has been based on empirical experience, where extensive numerical results showed that MVESbased algorithms may identify the underlying endmembers quite accurately under high signaltonoise ratios and without pure pixels. In this paper, we report some theoretical results on the endmember identifiability of the MVES criterion in the noiseless case. We employ an assumption that is a twomixture generalization of the purepixel assumption; particularly, we require a set of pixels, each being constituted by only two endmembers (rather than one as in the purepixel assumption), to exist in the data set. Under this assumption and some rather mild condition, we show that the MVES solution perfectly identifies the true endmembers. Numerical simulation results are provided to verify our theoretical results. Index Terms — Hyperspectral unmixing, minimum volume enclosing simplex, identifiability analysis, convex geometry