The Gaussian Transform of Distributions: Definition, Computation and Application

Abstract

Abstract-This paper introduces the general purpose Gaussian Transform of distributions, which aims at representing a generic symmetric distribution as an infinite mixture of Gaussian distributions. We start by the mathematical formulation of the problem and continue with the investigation of the conditions of existence of such a transform. Our analysis leads to the derivation of analytical and numerical tools for the computation of the Gaussian Transform, mainly based on the Laplace and Fourier transforms, as well as of the afferent properties set (e.g. the transform of sums of independent variables). The Gaussian Transform of distributions is then analytically derived for the Gaussian and Laplacian distributions, and obtained numerically for the Generalized Gaussian and the Generalized Cauchy distribution families. In order to illustrate the usage of the proposed transform we further show how an infinite mixture of Gaussians model can be used to estimate/denoise non-Gaussian data with linear estimators based on the Wiener filter. The decomposition of the data into Gaussian components is straightforwardly computed with the Gaussian Transform, previously derived. The estimation is then based on a two-step procedure, the first step consisting in variance estimation, and the second step in data estimation through Wiener filtering. To this purpose we propose new generic variance estimators based on the Infinite Mixture of Gaussians prior. It is shown that the proposed estimators compare favorably in terms of distortion with the shrinkage denoising technique, and that the distortion lower bound under this framework is lower than the classical MMSE bound.Index Terms-Gaussian mixture, Gaussian Transform of distributions, Generalized Gaussian, denoising, shrinkage I. INTRODUCTIONGaussian distributions are extensively used in the (broad sense) signal processing community, mainly for computational benefits. For instance, in an estimation problem Gaussian priors yield quadratic functionals and linear solutions. In ratedistortion and coding theories, closed form results are mostly available for Gaussian source and channel descriptions [11]. However, real data is most often non-Gaussian distributed, and is best described by other types of distribution (e.g in image processing, the most commonly used model for the wavelet coefficients distribution is the Generalized Gaussian distribution [6]). The goal of the work presented in this paper is to describe non-Gaussian distributions as an infinite mixture of Gaussian distributions, through direct computation of the mixing functions from the non-Gaussian distribution.In a related work [4], it was proven that any distribution can be approximated through a mixture of Gaussian up to an arbitrary level of precision. However, no hint was given by the author on how to obtain the desired mixture in the general case. In the radar community the problem of modeling nonGaussian radar clutters led to the theory of Spherically Invariant Random Processes (SIRP) [13], which aimed at generating n...

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