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Synthetic aperture radar (SAR) images are inherently affected by multiplicative speckle noise, which is due to the coherent nature of the scattering phenomenon. It appears sensible to reduce speckle in SAR images, provided that the structural features and textural information are not lost. We present a novel speckle removal algorithm within the framework of wavelet analysis. First, we show that the subband decompositions of logarithmically transformed SAR images are best described by alpha-stable distributions, a family of heavy-tailed densities. Consequently, we design a maximum a posteriori (MAP) estimator that exploits this a priori information. We use the alpha-stable model to develop a blind speckle-suppression processor that performs a non-linear operation on the data, and we relate this non-linearity to the degree of non-Gaussianity of the data. Finally, we compare our proposed method to a current state-of-the-art soft thresholding technique applied on an aerial image and we quantify the achieved performance improvement.