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.