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

where

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