Hyperspectral unmixing from a convex analysis and optimization perspective

In hyperspectral remote sensing, unmixing a data cube into spectral signatures and their corresponding abundance fractions plays a crucial role in analyzing the mineralogical composition of a solid surface. This paper describes a convex analysis perspective to (unsupervised) hyperspectral unmixing. Such an endeavor is not only motivated by the recent prevalence of convex optimization in signal processing, but also by the nature of hyperspectral unmixing (specifically, non-negativity and full additivity of abundances) that makes convex analysis a very suitable tool. By the notion of convex analysis, we formulate two optimization problems for solving hyperspectral unmixing, which have the intuitive ideas following the works by Craig and Winter respectively but adopt an optimization treatment different from those previous works. We show the connection of the two hyperspectral unmixing optimization problems, by proving that their optimal solutions become identical when pure pixels exist in the data. We also illustrate how the two problems can be conveniently handled by alternating linear programming. Monte Carlo simulations are presented to demonstrate the efficacy of the two hyperspectral unmixing formulations.