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Solving the inverse problem of compressive sensing in the context of single measurement vector (SMV) problem with an unknown block-sparsity structure is considered. For this purpose, we propose a sparse Bayesian learning (SBL) algorithm simplified via the approximate message passing (AMP) framework. In order to encourage the block-sparsity structure, we incorporate the concept of total variation, called Sigma-Delta, as a measure of block-sparsity on the support set of the solution. The AMP framework reduces the computational load of the proposed SBL algorithm and as a result makes it faster compared to the message passing framework. Furthermore, in terms of the mean-squared error between the true and the reconstructed solution, the algorithm demonstrates an encouraging improvement compared to the other algorithms.