New Theory for Unmixing ILL-Conditioned Hyperspectral Mixtures

Hyperspectral unmixing (HU), a blind source separation problem, aims at unambiguiously identifying the spectral signatures of the materials, as well as their abundances, from the measured hyperspectral mixtures. In real hyperspectral scenes, high correlation between the spectral signatures is commonly observed, making HU quite challenging. Although such ill-conditioning is critical for effective HU, it is often ignored in existing HU literature. To the best of our knowledge, existing preconditioning techniques, for reducing the condition number of the signature matrix, were developed based on the pure-pixel assumption, which can, however, be seriously violated in remote sensing. Under a relaxed purity assumption, with respect to the pure-pixel one, this paper proposes novel theory for unmixing ill-conditioned hyperspectral mixtures. Specifically, we exactly identify the John's ellipsoid (i.e., the maximum ellipsoid inscribed in the convex hull of the hyperspectral data vectors) via split augmented Lagrangian shrinkage algorithm (SALSA), and transform this ellipsoid into an Euclidean ball. This transformation brings the data vectors into a new space wherein the corresponding material signature vectors form a regular simplex, which is a very strong prior information. Based on this prior, we design an HU criterion, and prove its perfect identifiability under a very mild sufficient condition. Then, we demonstrate the feasibility of realizing our criterion via non-convex optimization and guarantee a stationary point solution.