Bayesian algorithms for supervised, semi-supervides, and unsupervised unmixing of hyperspectral images

This article describes fully Bayesian algorithms to unmix hyperspectral images. Each pixel of the hyperspectral image is decomposed as a combination of pure endmember spectra according to the linear mixing model. In a supervised context, the endmembers are assumed to be known. The unmixing problem consists of estimating the mixing coefficients under positivity and additivity constraints. An appropriate distribution is chosen as prior distribution for these coefficients, that are estimated from their posterior distribution. A Markov chain Monte Carlo (MCMC) algorithm is developed to approximate the estimators. In a semi-supervised framework, the spectra involved in the mixtures are assumed to be unknown. They are supposed to belong to a known spectral library. A reversible-jump MCMC algorithm allows one to solve the resulting model selection problem. Finally, in a final step, the previous algorithms are extended to handle the unsupervised unmixing problem, i.e., to estimate the endmembers and the mixing coefficients jointly. This blind source separation problem is solved in a lower-dimensional space, which effectively reduces the number of degrees of freedom of the unknown parameters.