Generative Simplex Mapping: Non-Linear Endmember Extraction and Spectral Unmixing for Hyperspectral Imagery

Abstract

We introduce a new model for non-linear endmember extraction and spectral unmixing of hyperspectral imagery called Generative Simplex Mapping (GSM). The model represents endmember mixing using a latent space of points sampled within a (n−1)-simplex corresponding to n unique sources. Barycentric coordinates within this simplex are naturally interpreted as relative endmember abundances satisfying both the abundance sum-to-one and abundance non-negativity constraints. Points in this latent space are mapped to reflectance spectra via a flexible function combining linear and non-linear mixing. Due to the probabilistic formulation of the GSM, spectral variability is also estimated by a precision parameter describing the distribution of observed spectra. Model parameters are determined using a generalized expectation-maximization algorithm, which guarantees non-negativity for extracted endmembers. We first compare the GSM against three varieties of non-negative matrix factorization (NMF) on a synthetic data set of linearly mixed spectra from the USGS spectral database. Here, the GSM performed favorably for both endmember accuracy and abundance estimation with all non-linear contributions driven to zero by the fitting procedure. In a second experiment, we apply the GTM to model non-linear mixing in real hyperspectral imagery captured over a pond in North Texas. The model accurately identified spectral signatures corresponding to near-shore algae, water, and rhodamine tracer dye introduced into the pond to simulate water contamination by a localized source. Abundance maps generated using the GSM accurately track the evolution of the dye plume as it mixes into the surrounding water.