Gibbs Random Fields : Temperature and Parameter Analysis

Gibbs random eld (GRF) models work well for synthesizing complex natural-looking image data with a small number of parameters; however, estimation methods for these parameters have a lot of problems. This paper addresses the analysis problem in a new way by examining the role of the temperature parameter of the Gibbs distribution. Studies of the model energy with respect to the temperature are used to indicate pattern equilibrium and regions of diierent behavior , analogous to the existence of distinct phases in a physical system. The results on equilibrium and regions of diierent \phases" are ooered as explanations for some of the peculiar behavior of current estimation algorithms. 1 Gibbs random elds This paper focuses on the discrete Gibbs random eld (GRF), deened as follows. Let an image be represented by a nite rectangular M N lattice S with a neighborhood structure N = fNs;s 2 Sg where Ns S is the set of sites which are neighbors of the site s 2 S. Every site has a graylevel value xs 2 = f0; 1; : : : ; n ? 1g. Let x be the vector (xs; 1 s jSj) of site graylevel values and be the set of all conngurations taken by x. A neighborhood structure is said to be symmetric if 8s; r 2 S, s 2 Nr if and only if r 2 Ns. For the nite periodic lattice S with a symmetric neighborhood structure, one can deene a Gibbs energy. There are many ways to deene the energy ; the choice studied here is the auto-binomial energy of 1], which has been shown to synthesize a variety of natural looking image textures 2]. The homogeneous auto-binomial energy is E(x) = ? X s2S xs + X r2Ns rxsxr ! ; (1) where the model parameters are , the external eld, and r, the bonding interactions. A joint probability distribution is assigned to the Gibbs energy yielding the Gibbs random eld, P(x) = 1 Z exp ? 1 T E(x) ; (2) where Z is a positive normalizing constant known in the physics literature as a partition function and T is the \temperature" of the eld. The synthesis process is one of nding the con-guration in which maximizes the probability P(x), minimizing the Gibbs energy. Image data is typically synthesized iteratively, using a Monte Carlo method such as the Metropolis exchange algorithm. In this algorithm it …