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The spatial property can be modelled through different aspects,
among which, the contextual constraint is a general and
powerful one. Markov random field (MRF) theory provides a
convenient and consistent way to model contextdependent entities
such as image pixels and correlated features. This is achieved by
characterizing mutual influences among such entities using
conditional MRF distributions.
In an MRF, the sites in
are related to one another
via a neighbourhood system, which is defined as
,
where
is the set of sites neighbouring i,
and
.
A random field X said to be an MRF on
with
respect to a neighbourhood system
if and only if
Note, the neighbourhood system can be multidimensional. According
to the HammersleyClifford theorem [1], an MRF can
equivalently be characterized by a Gibbs distribution. Thus,

(5) 
where

(6) 
is a normalizing constant called the partition function,
and
U(x) is an energy function of the form

(7) 
which is a sum of clique potentials
V_{c}(x) over
all possible cliques
.
A clique c is defined
as a subset of sites in
in which every pair of
distinct sites are neighbours, except for singlesite cliques. The
value of
V_{c}(x) depends on the local configuration on
clique c. For more detail on MRF and Gibbs distribution see
[12].
Next: Hidden Markov Random Field
Up: Hidden Markov Random Field
Previous: Finite Mixture Model
Yongyue Zhang
20000511