A novel speckle suppression method for medical ultrasound images is presented. First, the logarithmic transform of the original image is analyzed into the multiscale wavelet domain. The authors show that the subband decompositions of ultrasound images have significantly non-Gaussian statistics that are best described by families of heavy-tailed distributions such as the alpha-stable. Then, the authors design a Bayesian estimator that exploits these statistics. They use the alpha-stable model to develop a blind noise-removal processor that performs a nonlinear operation on the data. Finally, the authors compare their technique with current state-of-the-art soft and hard thresholding methods applied on actual ultrasound medical images and they quantify the achieved performance improvement.