Density Modeling of Images using a Generalized Normalization Transformation

We introduce a parametric nonlinear transformation that is well-suited for Gaussianizing data from natural images. After a linear transformation of the data, each component is normalized by a pooled activity measure, computed by exponentiating a weighted sum of rectified and exponentiated components and an additive constant. We optimize the parameters of this transformation (linear transform, exponents, weights, constant) over a database of natural images, directly minimizing the negentropy of the responses. We find that the optimized transformation successfully Gaussianizes the data, achieving a significantly smaller mutual information between transformed components than previous methods including ICA and radial Gaussianization. The transformation is differentiable and can be efficiently inverted, and thus induces a density model on images. We show that samples of this model are visually similar to samples of natural image patches. We also demonstrate the use of the model as a prior density in removing additive noise. Finally, we show that the transformation can be cascaded, with each layer optimized (unsupervised) using the same Gaussianization objective, to capture additional probabilistic structure.