3 S.D. Babacan R. Molina A.K. Katsaggelos 2009 IEEE International Conf. on Acoustics, Speech, and Signal Processing (ICASSP’09) Taipei, Taiwan 19/04/2009 In this paper we model the components of the compressive sensing (CS) problem using the Bayesian framework by utilizing a hierarchical form of the Laplace prior to model sparsity of the unknown signal. This signal prior includes some of the existing models as special cases and achieves a high degree of sparsity. We develop a constructive (greedy) algorithm resulting from this formulation where necessary parameters are estimated solely from the observation and therefore no user-intervention is needed. We provide experimental results with synthetic 1D signals and images, and compare with the state-of-the-art CS reconstruction algorithms demonstrating the superior performance of the proposed approach.