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A nonparametric Bayesian estimator in the wavelet domain is presented. In this approach, we propose a prior model based on the α-stable densities to capture the sparseness of the wavelet coefficients. An attempt to apply this model image wavelet-denoising have been already proposed in A.Achim et al. (2001). However, despite its efficacy in modeling the heavy-tail behaviour of the empirical detail coefficients densities, their denoiser proves very poor in practice and suffers from many drawbacks such as the weakness of the hyperparameters estimator associated with the α-stable prior. Here, we propose to overcome these limitations using the scale-mixture of Gaussians as an analytical approximation for α-stable densities. Exploiting this prior, we design a Bayesian L2-loss nonlinear denoiser.