A Distance Geometric Framework for Nonlinear Hyperspectral Unmixing

In this article, a distance geometry-based framework for hyperspectral image unmixing is presented. A manifold representation of the data set is generated by creation of a nearest-neighbor graph on which shortest paths are calculated yielding a geodesic distance matrix. Instead of unfolding the manifold in a lower-dimensional Euclidean space, it is proposed to work directly on the manifold. To do so, algorithms need to be rewritten in terms of distance geometry. Building further on earlier work, where distance-based dimensionality estimation and endmember extraction methods were presented, we will propose a distance geometric version of the actual unmixing (abundance estimation) step. In this way, a complete distance geometric unmixing framework is obtained that is efficient compared to the classical methods based on optimization. Furthermore, the distance geometry-adapted algorithms can be applied on nonlinear data manifolds by employing geodesic distances. In the experiments, we demonstrate this by comparing the obtained nonlinear framework to its linear counterpart.