Bayesian Compressive Sensing Using Non-Convex Priors

In this work we developed a novel Bayesian formulation for the reconstruction from compressive measurements. We demonstrate that non-convex sparsity priors based on l_p-norms, with p smaller or equal to 1, can be employed within a Bayesian framework by utilizing majorization-minimization methods. Non-convex priors enforce a high level of sparsity, and they are therefore very suitable for the compressive sensing recovery. By utilizing a fully Bayesian analysis of the compressive sensing system and employing a variational Bayesian analysis for inference, the proposed framework provides certain advantages, such as (i) the model parameters (such as observation noise variance) are estimated along with the unknown signal, leaving no free parameters to tune; (ii) the distributions of the unknowns are estimated, which can be used to calculate the uncertainty of the estimates. We also show that some existing methods can be derived as special cases of the proposed framework. Experimental results demonstrate the advantage of non-convex priors and the high performance of the proposed algorithm compared to existing algorithms commonly used for compressive sensing recovery.

This work appeared in European Signal Processing Conference (EUSIPCO’09).