An Investigation of Likelihoods and Priors for Bayesian Endmember Estimation

A Gibbs sampler for piece-wise convex hyperspectral unmixing and endmember detection is presented. The standard linear mixing model used for hyperspectral unmixing assumes that hyperspectral data reside in a single convex region. However, hyperspectral data is often nonconvex. Furthermore, in standard unmixing methods, endmembers are generally represented as a single point in the high dimensional space. However, the spectral signature for a material varies as a function of the inherent variability of the material or environmental conditions. Therefore, it is more appropriate to represent each endmember as a full distribution to incorporate the variability and utilize this information during spectral unmixing. A Gibbs sampler that searches for several sets of endmember distributions, i.e. a piece-wise convex representation, is presented. The hyperspectral data is partitioned among the sets of endmember distributions using a Dirichlet process prior that also estimates the number of needed sets. The proposed likelihood follows from a convex combination of normal endmember distributions with a Dirichlet prior on the abundance values. A normal distribution is also applied as a prior for the mean values of the endmember distributions. The Gibbs sampler that is presented partitions the data into convex regions, determines the number of convex regions required and determines endmember distributions and abundance values for all convex regions. Results are presented on hyperspectral data that indicate the ability of the method to effectively estimate endmember distributions and the number of sets of endmember distributions.