A Gaussian mixture model representation of endmember variability for spectral unmixing

Endmember variability complicates the problem of spectral unmixing. This variability is typically represented by probability distributions or spectral libraries. The present work describes a new distributional representation based on Gaussian Mixture Models (GMMs). The most common form in this setting is the Normal Compositional Model (NCM), wherein the endmembers for each pixel are modeled as samples drawn from unimodal Gaussians. In reality, however, the distribution of spectra from a material may be multi-modal. We first show that a linear combination of GMM random variables is also a GMM. This allows us to probabilistically formulate hyperspectral pixel likelihoods as combinations of independent endmember random variables. Then, after adding a reasonable smoothness and sparsity prior on the abundances, we obtain a standard Bayesian maximum a posteriori (MAP) problem for abundance and endmember parameter estimation. A generalized expectation-maximization (EM) algorithm is used to minimize the MAP objective function. We tested the GMM approach on two real datasets, and showcased its efficacy for modeling endmember variability by comparing it to current popular methods.