This paper revisits blind source separation of instantaneously mixed quasi-stationary sources (BSS-QSS), 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 so-called 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 BSS-QSS context, an algebraic preprocessing procedure is proposed to suppress short-term source cross-correlation 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 BSS-QSS, 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.

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 MVES-based algorithms may identify the underlying endmem-bers quite accurately under high signal-to-noise 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 two-mixture generaliza-tion of the pure-pixel assumption; particularly, we require a set of pixels, each being constituted by only two endmembers (rather than one as in the pure-pixel 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. Numeri-cal simulation results are provided to verify our theoretical results. Index Terms — Hyperspectral unmixing, minimum volume en-closing simplex, identifiability analysis, convex geometry