In compressive sensing (CS) the known structure in the transform coefficients may be leveraged to improve reconstruction accuracy. We here develop a hierarchical statistical model applicable to both wavelet and JPEG-based DCT bases, in which the tree structure in the sparseness pattern is exploited explicitly. The analysis is performed efficiently via variational Bayesian (VB) analysis, and comparisons are made with MCMC-based inference, and with many of the CS algorithms in the literature. Performance is assessed for both noise-free and noisy CS measurements, based on both JPEG-DCT and wavelet representations. © 2009 IEEE.