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Bayesian compressive sensing (CS) is considered for signals and images that are sparse in a wavelet basis. The statistical structure of the wavelet coefficients is exploited explicitly in the proposed model, and, therefore, this framework goes beyond simply assuming that the data are compressible in a wavelet basis. The structure exploited within the wavelet coefficients is consistent with that used in wavelet-based compression algorithms. A hierarchical Bayesian model is constituted, with efficient inference via Markov chain Monte Carlo (MCMC) sampling. The algorithm is fully developed and demonstrated using several natural images, with performance comparisons to many state-of-the-art compressive-sensing inversion algorithms.