Compressive Blind Source Separation

The central goal of compressive sensing is to reconstruct a signal that is sparse or compressible in some basis using very few measurements. However reconstruction is often not the ultimate goal and it is of considerable interest to be able to deduce attributes of the signal from the measurements without explicitly reconstructing the full signal. This paper solves the blind source separation problem not in the high dimensional data domain, but in the low dimensional measurement domain. It develops a Bayesian inference framework that integrates hidden Markov models for sources with compressive measurement. Posterior probabilities are calculated using a Markov Chain Monte Carlo (MCMC) algorithm. Simulation results are provided for one-dimensional signals and for two-dimensional images, where hidden Markov tree models of the wavelet coefficients are considered. The integrated Bayesian framework is shown to outperform standard approaches where the mixtures are separated in the data domain.