Modeling Image Structure with Factorized Phase-Coupled Boltzmann Machines

TL;DR

This paper introduces a factorized phase-coupled Boltzmann machine that models local amplitude and phase structure in natural images, capturing higher-order dependencies and phase relationships.

Contribution

It extends previous models to include phase coupling in 2D subspaces, enabling better representation of local image structure and phase dependencies.

Findings

Learned subspaces resemble Gabor filters

Model captures phase dependencies in natural images

Enhances understanding of image local structure

Abstract

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 (Ranzato and Hinton, 2010). Here, we extend this model to $L_p$-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 $L_2$-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.