Hierarchical Bayesian sparse source separation of hyperspectral signals

We consider the separation of signals having a simplex constraint based on observations of noisy data using a fully Bayesian result. A joint density for the abundances (the simplex-constrained signal) is established. To specifically allow for the abundances to be sparse, this joint density is endowed with a parameter which is parameterized to encourage sparseness, unlike related previous work which did not specifically impose the sparsity assumption. Overall, a Gibbs sampling framework is established from which all parameters can be learned. A Metropolis sampling framework is developed for the abundances (mixture coefficients) which explicitly and efficiently represents the sparseness. For the parameter of the Dirichlet distribution governing sparseness, the posterior is shown to be closely approximated by a Gamma. Thus the entire set of parameters can be efficiently learned by sampling.