Hyperspectral unmixing with projection onto convex sets using distance geometry

In this paper, a new method is presented to solve the spectral unmixing problem. The method is based on the projection on convex sets principle, in which a simplex is considered as an intersection of a plane and half-spaces, and the abundances are obtained by alternatively projecting data onto the half-spaces using the well-known Dykstra algorithm. In this paper, every step of such a recently developed alternating projection unmixing algorithm is rephrased using distance geometry, i.e. using only the spectral distances between the data points and the endmembers. This distance geometric approach allows to use any distance metric other than the Euclidean one. The experimental validation shows that the method provides exact results for the fully constrained unmixing problem. Moreover, we demonstrate the usefulness of the method for nonlinear unmixing, using geodesic distances on the data manifold.