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Normalized and Canonical Forms

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).