A closed-form nonparametric Bayesian estimator in the wavelet domain of images using an approximate α-stable prior

In this paper, a nonparametric Bayesian estimator in the wavelet domains is presented. In this approach, we propose a prior statistical model based on the α-stable densities adapted to capture the sparseness of the wavelet detail coefficients. An attempt to apply this model in the context of wavelet denoising have been already proposed in (Achim, A., Bezerianos, A., Tsakalides, P., 2001. Novel Bayesian multiscale method for speckle removal in medical ultrasound images. IEEE Trans. Med. Imag. 20, 772–783). However, despite its efficacy in modeling the heavy tail behavior of the empirical wavelet coefficients histograms, their denoiser proves very poor in practice especially at low SNRs. It suffers from many drawbacks such as numerical instability because of the lack of a closed-form expression of the Bayesian shrinkage rule, and the weakness of the estimator of the hyperparameters associated with the α-stable prior. Here, we propose to overcome these limitations using the scale mixture of Gaussians theorem as an analytical approximation for α-stable densities, which is not known in general, in order to obtain a closed-form expression of our Bayesian denoiser.