On the endmember identifiability of Craig´s criterion for hyperspectral unmixing: A statistical analysis for three-source case

Hyperspectral unmixing (HU) is a process to extract the underlying endmember signatures (or simply endmembers) and the corresponding proportions (abundances) from the observed hyperspectral data cloud. The Craig´s criterion (minimum volume simplex enclosing the data cloud) and the Winter´s criterion (maximum volume simplex inside the data cloud) are widely used for HU. For perfect identifiability of the endmembers, we have recently shown in [1] that the presence of pure pixels (pixels fully contributed by a single endmember) for all endmembers is both necessary and sufficient condition for Winter´s criterion, and is a sufficient condition for Craig´s criterion. A necessary condition for endmember identifiability (EI) when using Craig´s criterion remains unsolved even for three-endmember case. In this work, considering a three-endmember scenario, we endeavor a statistical analysis to identify a necessary and statistically sufficient condition on the purity level (a measure of mixing levels of the endmembers) of the data, so that Craig´s criterion can guarantee perfect identification of endmembers. Precisely, we prove that a purity level strictly greater than 1/√(2) is necessary for EI, while the same is sufficient for EI with probability-1. Since the presence of pure pixels is a very strong requirement which is seldom true in practice, the results of this analysis foster the practical applicability of Craig´s criterion over Winter´s criterion, to real-world problems.