The G Factor: Relating Distributions on Features to Distributions on Images

Abstract

We describe the g-factor, which relates probability distributions on image features to distributions on the images themselves. The g-factor depends only on our choice of features and lattice quanti(cid:173) zation and is independent of the training image data. We illustrate the importance of the g-factor by analyzing how the parameters of Markov Random Field (i.e. Gibbs or log-linear) probability models of images are learned from data by maximum likelihood estimation. In particular, we study homogeneous MRF models which learn im(cid:173) age distributions in terms of clique potentials corresponding to fea(cid:173) ture histogram statistics (d. Minimax Entropy Learning (MEL) by Zhu, Wu and Mumford 1997 [11]) . We first use our analysis of the g-factor to determine when the clique potentials decouple for different features . Second, we show that clique potentials can be computed analytically by approximating the g-factor. Third, we demonstrate a connection between this approximation and the Generalized Iterative Scaling algorithm (GIS), due to Darroch and Ratcliff 1972 [2], for calculating potentials. This connection en(cid:173) ables us to use GIS to improve our multinomial approximation, using Bethe-Kikuchi[8] approximations to simplify the GIS proce(cid:173) dure. We support our analysis by computer simulations.