Modeling Image Structure with Factorized Phase-Coupled Boltzmann Machines

We describe a model for capturing the statistical structure of local amplitude and local spatial phase in natural images. The model is based on a recently developed, factorized third-order Boltzmann machine that was shown to be effective at capturing higher-order structure in images by modeling dependencies among squared filter outputs [1]. Here, we extend this model to Lp-spherically symmetric subspaces. In order to model local amplitude and phase structure in images, we focus on the case of two dimensional subspaces, and the L2-norm. When trained on natural images the model learns subspaces resembling quadrature-pair Gabor filters. We then introduce an additional set of hidden units that model the dependencies among subspace phases. These hidden units form a combinatorial mixture of phase coupling distributions, concentrated in the sum and difference of phase pairs. When adapted to natural images, these distributions capture local spatial phase structure in natural images.