## Abstract

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 technique known as the mean-field approximation, which is widely used in statistical physics. Our mean-field theory is general in that it is valid for any number of gray levels, any pairwise 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 of 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 gain insight into the Gibbs model behavior in the neighborhood of these temperatures. These analytical results are verified by computer simulations that use a novel mean-field descent algorithm. An important spinoff of our mean-field theory is that it allows us to compute approximations for the correlation functions of GRF models, thus bridging the gap between neighborhood-based and correlation-based a priori image models.

Original language | British English |
---|---|

Pages (from-to) | 167-186 |

Number of pages | 20 |

Journal | Journal of Mathematical Imaging and Vision |

Volume | 3 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1993 |

## Keywords

- critical temperatures
- Gibbs-Markov random fields
- image modeling
- mean-field theory
- textures