Mean-field phase transitions for Gibbs random fields

The use of Gibbs random fields (GRF) to model images poses the important problem of the dependence of the patterns sampled from the Gibbs distribution on its parameters. Sudden changes in these patterns as the parameters are varied are known as phase transitions. In this paper, we concentrate on developing a general deterministic theory for the study of phase transitions when a single parameter, namely the temperature, is varied. This deterministic framework is based on a widely used technique in statistical physics known as the mean-field approximation. Our mean-field theory is general in that it is valid for any number of graylevels, any interaction potential, any neighborhood structure or size, and any set of constraints imposed on the desired images. The mean-field approximation is used to compute closed-form estimates for the critical temperatures at which phase transitions occur for two texture models widely used in the image modeling literature: the Potts model and the autobinomial model. The mean-field model allows us to predict the pattern structure in the neighborhood of these temperatures. These analytical results are verified by computer simulations using a novel mean-field descent algorithm.