1.2.5

It is known that the choices of clique potential functions for a
specific MRF is not unique; there may exist many equivalent choices
which specify the same Gibbs distribution. However, there exists a
unique normalized potential, called the * canonical potential*, for
every MRF [Griffeath 1976].

Let be a countable label set. A clique potential function
is said to be * normalized* if
whenever for some , takes a particular value in
. The particular value can be any element in , * e.g. *
0 in
. Griffeath (1976)
establishes the mathematical relationship between an MRF distribution
and the unique canonical representation of clique potentials
in the corresponding Gibbs distribution [Griffeath 1976 ; Kindermann and Snell 1980]. The result is described
below.

Let **F** be a random field on a finite set with local
characteristics .
Then **F** is a Gibbs field with * canonical potential function*
defined by the following:

where denotes the empty set, **|c-b|** is the number of elements
in the set **c-b** and

is the configuration which agrees with **f** on set **b** but assigns the
value **0** to all sites outside of **b**. For nonempty **c**, the potential
can also be obtained as

where **i** is any element in **b**. Such canonical potential function is
* unique* for the corresponding MRF. Using this result, the canonical
can be computed if is known.

However, in MRF modeling using Gibbs distributions, is defined after is determined and therefore, it is difficult to compute the canonical from directly. Nonetheless, there is an indirect way: Use a non-canonical representation to derive and then canonicalize it using Griffeath's result to obtain the unique canonical representation.

The normalized potential functions appear to be immediately useful. For instance, for the sake of economy, one would use the minimal number of clique potentials or parameters to represent an MRF for a given neighborhood system. The concept of normalized potential functions can be used to reduce the number of nonzero clique parameters (see Chapter 6).