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
