Variational Bayesian Learning of SMoGs: Modelling and Their Application to Synthetic Aperture Radar

We show how modern Bayesian Machine Learning tools can be effectively used in order to develop efficient methods for filtering Earth Observation signals. Bayesian statistical methods can be thought of as a generalization of the classical least-squares adjustment methods where both the unknown signals and the parameters are endowed with probability distributions, the priors. Statistical inference under this scheme is the derivation of posterior distributions, that is, distributions of the unknowns after the model has seen the data. Least squares can then be thought of as a special case that uses Gaussian likelihoods, or error statistics. In principle, for most non-trivial models, this framework requires performing integration in high-dimensional spaces. Variational methods are effective tools for approximate inference in Statistical Machine Learning and Computational Statistics. In this paper, after introducing the general variational Bayesian learning method, we apply it to the modelling and implementation of sparse mixtures of Gaussians (SMoG) models, intended to be used as adaptive priors for the efficient representation of sparse signals in applications such as wavelet-type analysis. Wavelet decomposition methods have been very successful in denoising real-world, non-stationary signals that may also contain discontinuities. For this purpose we construct a constrained hierarchical Bayesian model capturing the salient characteristics of such sets of decomposition coefficients. We express our model as a Dirichlet mixture model. We then show how variational ideas can be used to derive efficient methods for bypassing the need for integration: the task of integration becomes one of optimization. We apply our SMoG implementation to the problem of denoising of Synthetic Aperture Radar images, inherently affected by speckle noise, and show that it achieves improved performance compared to established methods, both in terms of speckle reduction and image feature preservation.