A set of random variables F is said to be a Gibbs random field (GRF) on with respect to if and only if its configurations obey a Gibbs distribution. A Gibbs distribution takes the following form
is a normalizing constant called the partition function, T is a constant called the temperature which shall be assumed to be 1 unless otherwise stated, and is the energy function. The energy
is a sum of clique potentials over all possible cliques . The value of depends on the local configuration on the clique c. Obviously, the Gaussian distribution is a special member of this Gibbs distribution family.
A GRF is said to be homogeneous if is independent of the relative position of the clique c in . It is said to be isotropic if is independent of the orientation of c. It is considerably simpler to specify a GRF distribution if it is homogeneous or isotropic than one without such properties. The homogeneity is assumed in most MRF vision models for mathematical and computational convenience. The isotropy is a property of direction-independent blob-like regions.
To calculate a Gibbs distribution, it is necessary to evaluate the partition function Z which is the sum over all possible configurations in . Since there are a combinatorial number of elements in for a discrete , as illustrated in Section 1.1.2, the evaluation is prohibitive even for problems of moderate sizes. Several Approximation methods exist for solving this problem ( cf. Chapter 7).
measures the probability of the occurrence of a particular configuration, or ``pattern'', f. The more probable configurations are those with lower energies. The temperature T controls the sharpness of the distribution. When the temperature is high, all configurations tend to be equally distributed. Near the zero temperature, the distribution concentrates around the global energy minima. Given T and , we can generate a class of ``patterns'' by sampling the configuration space according to ; see Section 2.4.1.
For discrete labeling problems, a clique potential can be specified by a number of parameters. For example, letting be the local configuration on a triple-clique , takes a finite number of states and therefore takes a finite number of values. For continuous labeling problems, can vary continuously. In this case, is a (possibly piecewise) continuous function of .
Sometimes, it may be convenient to express the energy of a Gibbs distribution as the sum of several terms, each ascribed to cliques of a certain size, that is,
The above implies a homogeneous Gibbs distribution because , and are independent of the locations of i, and . For non-homogeneous Gibbs distributions, the clique functions should be written as , , and so on.
An important special case is when only cliques of size up to two are considered. In this case, the energy can also be written as
Note that in the second term on the RHS, and are two distinct cliques in because the sites in a clique are ordered. The conditional probability can be written as