Title: Nonlinear extraction of 'Independent Components' of
elliptically symmetric densities using radial Gaussianization


Authors: Siwei Lyu and Eero P. Simoncelli 


We consider the problem of efficiently encoding a signal by transforming it to a new
representation whose components are statistically independent (also known as factorial).
A widely studied family of solutions, generally known as 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 signal density is non-Gaussian but elliptically symmetric. In this case, no linear
transform suffices to properly decompose the signal into independent components,
and thus, the ICA methodology fails. We show that a simple nonlinear transformation,
which we call radial Gaussianization (RG), provides an exact solution for this case. We
then examine this methodology in the context of natural image statistics, demonstrating
that joint statistics of spatially proximal coefficients in a multi-scale image representation
are better described as elliptical than factorial. We quantify this by showing that
reduction in dependency achieved by RG is far greater than that achieved by ICA, for
local spatial neighborhoods. We also show that the RG transformation may be closely
approximated by divisive normalization transformations that have been used to model
the nonlinear response properties of visual neurons, and that have been shown to reduce
dependencies between multi-scale image coefficients.