Unilateral approximation of Gibbs random field images

We exploit the relationship between a general Gibbs random field and a mutually compatible Gibbs random field image, we investigate the possibility of approximating a general Gibbs random field by a mutually compatible Gibbs random field, and we study the consequences of such an approximation. This study is important, since, in some cases, various problems related to the modeling of images via Gibbs random fields may be simplified by restricting our interest to a special and convenient class of Gibbs random fields, the class of mutually compatible Gibbs random fields. A general analysis is presented, which results in some interesting conclusions about the desired relationship. We consider two approaches to the approximation problem. The first approach, which is more accurate, can be easily applied to the case of binary images. For nonbinary images this approach becomes cumbersome, since it requires the solution of a large system of nonlinear equations. The second approach, which is simpler but less accurate, is based on the direct relationship between the parameters of a general Gibbs random field and the parameters of a mutually compatible Gibbs random field. Simulation examples demonstrate various aspects of our analysis.