Random cascades of Gaussian scale mixtures and their use in modeling natural images with application to denoising

Multiresolution representations play an important role in image processing and computer vision, as well as in modeling stochastic processes. We have developed a semi-parametric class of non-Gaussian multiscale statistical processes defined by random cascades on wavelet trees. This model class is rich enough to accurately capture the remarkably regular non-Gaussian features of natural images, but sufficiently structured to permit estimation of the underlying state variables. We showed that our models accurately fit both the marginal and joint histograms of wavelet coefficients from natural images. We developed a Newton-like method for exact MAP state estimation that exploits fast algorithms for tree estimation, and hence is very efficient. Applications of this algorithm to denoising of both 1D signals and natural images were presented. The GSM-tree model class is related to a number of previous approaches to image coding and denoising.