The g Factor: Relating Distributions on Features to Distributions on Images
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 quantization 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 image distributions in terms of clique potentials corresponding to feature histogram statistics (cf. 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 enables us to use GIS to improve our multinomial approximation, using Bethe-Kikuchi[8] approximations to simplify the GIS procedure. We support our analysis by computer simulations.