MCMC Algorithms for Supervised and Unsupervised Linear Unmixing of Hyperspectral Images

¹ University of Toulouse, IRIT/INP-ENSEEIHT, 2 rue Camichel, BP. 7122, 31071 Toulouse Cedex 7, France
² IRCCyN - CNRS UMR 6597, ECN, 1 rue de la Noë, BP. 92101, 44321 Nantes Cedex 3, France
³ University of Michigan, Department of EECS, 1301 Beal Avenue, Ann Arbor, 48109-2122, USA

Abstract

In this paper, we describe two fully Bayesian algorithms that have been previously proposed to unmix hyperspectral images. These algorithms relies on the widely admitted linear mixing model, i.e. each pixel of the hyperspectral image is decomposed as a linear combination of pure endmember spectra. First, the unmixing problem is addressed in a supervised framework, i.e., when the endmembers are perfectly known, or previously identified by an endmember extraction algorithm. In such scenario, the unmixing problem consists of estimating the mixing coefficients under positivity and additivity constraints. Then the previous algorithm is 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. For both scenarios, appropriate distributions are assigned to the unknown parameters, that are estimated from their posterior distribution. Markov chain Monte Carlo (MCMC) algorithms are then developed to approximate the Bayesian estimators.