On Using Projection Onto Convex Sets for Solving the Hyperspectral Unmixing Problem

Abstract

An important problem in hyperspectral unmixing is solving the inversion problem, which determines the abundances of each endmember in each pixel, taking the constraints on these abundances into account. In this paper, we present a new geometrical method for solving this inversion problem, based on the equivalence with the simplex projection problem, and projection onto convex sets. By writing the simplex as an intersection of a plane and convex halfspaces, an alternating projection algorithm is constructed based on the Dykstra algorithm. We show that the resulting algorithm can be used to successfully solve the spectral unmixing problem, and yields results that are comparable to those obtained with state-of-the-art methods. The runtime required is very competitive, and the very simple nature of the algorithm allows for highly efficient implementations.