Calculation of Geodesic Distances in Nonlinear Mixing Models: Application to the Generalized Bilinear Model

Abstract

Recently, several nonlinear techniques have been proposed in hyperspectral image processing for classification and unmixing applications. A popular data-driven approach for treating nonlinear problems employs the geodesic distances on the data manifold as property of interest. These geodesic distances are approximated by the shortest path distances in a nearest neighbor graph constructed in the data cloud. Although this approach often works well in practical applications, the graph-based approximation of these geodesic distances often fails to capture correctly the true nonlinear structure of the manifold, causing deviations in the subsequent algorithms. On the other hand, several model-based nonlinear techniques have been introduced as well and have the advantage that one can, in theory, calculate the geodesic distances analytically. In this letter, we demonstrate how one can calculate the true geodesics, and their lengths, on any manifold induced by a nonlinear hyperspectral mixing model. We introduce the required techniques from differential geometry, show how the constraints on the abundances can be integrated in these techniques, and present a numerical method for finding a solution of the geodesic equations. We demonstrate this technique on the recently developed generalized bilinear model, which is a flexible model for the nonlinearities introduced by secondary reflections. As an application of the technique, we demonstrate that multidimensional scaling applied to these geodesic distances can be used as a preprocessing step to linear unmixing, yielding better unmixing results on nonlinear data when compared to principal component analysis and outperforming ISOMAP.