We consider the problem of efficiently encoding a signal by transforming it to a new representation whose components are statistically independent. A widely studied linear solution, independent components analysis (ICA), exists for the case when the signal is generated as a linear transformation of independent non- Gaussian sources. Here, we examine a complementary case, in which the source is non-Gaussian but elliptically symmetric. In this case, no linear transform suffices to properly decompose the signal into independent components, but we show that a simple nonlinear transformation, which we call radial Gaussianization (RG), is able to remove all dependencies. We then demonstrate this methodology in the context of natural signal statistics. We first show that the joint distributions of bandpass filter responses, for both sound and images, are better described as elliptical than linearly transformed independent sources. Consistent with this, we demonstrate that the reduction in dependency achieved by applying RG to either pairs or blocks of bandpass filter responses is significantly greater than that achieved by PCA or ICA.