  Most signal processing problems involve the challenging task of
multidimensional probability density function (PDF) estimation. In this work,
we propose a solution to this problem by using a family of Rotation-based
Iterative Gaussianization (RBIG) transforms. The general framework consists of
the sequential application of a univariate marginal Gaussianization transform
followed by an orthonormal transform. The proposed procedure looks for
differentiable transforms to a known PDF so that the unknown PDF can be
estimated at any point of the original domain. In particular, we aim at a zero
mean unit covariance Gaussian for convenience. RBIG is formally similar to
classical iterative Projection Pursuit (PP) algorithms. However, we show that,
unlike in PP methods, the particular class of rotations used has no special
qualitative relevance in this context, since looking for interestingness is not
a critical issue for PDF estimation. The key difference is that our approach
focuses on the univariate part (marginal Gaussianization) of the problem rather
than on the multivariate part (rotation). This difference implies that one may
select the most convenient rotation suited to each practical application. The
differentiability, invertibility and convergence of RBIG are theoretically and
experimentally analyzed. Relation to other methods, such as Radial
Gaussianization (RG), one-class support vector domain description (SVDD), and
deep neural networks (DNN) is also pointed out. The practical performance of
RBIG is successfully illustrated in a number of multidimensional problems such
as image synthesis, classification, denoising, and multi-information
estimation.
