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is called a random field on
. The joint probability of the random field taking a
particular combination of values is denoted
P
(
S
).
A Markov Random Field is defined as a random field in which the proba-
bility
P
(
f
i
) is only dependent on
f
i
f
and some of its neighbors. Therefore, a
N
neighborhood system
is defined as
N
=
{
N
i
|
i
∈
S
}
(1)
N
i
is the index set of sites neighboring
i
. The neighboring relationship
has the following properties:
where
1. a site is not a neighbor of itself:
i
∈
N
i
,
2. the neighboring relationship is mutual:
i
i
∈
N
i
.
For a regular lattice
S
, the neighboring set of
i
is usually defined as the set of
sites within a radius of
i
. Note that sites at or near the boundary of the lattice
have fewer neighbors. The Markovianity constraint is then expressed by
∈
N
i
⇐⇒
P
(
f
i
|
f
−{
f
i
}
)=
P
(
f
i
|
f
N
i
)
(2)
where
f
−{
f
i
}
denotes all values of the random field except for
f
i
itself, and
i
∈
N
i
}
f
N
i
=
stands for the labels at the sites neighbouring
i
.
Let us construct a graph on
{
f
i
|
in which the edges represent the neighboring
relationships. Now consider the cliques in this graph. A clique is a subset of
vertices so that every two vertices are connected by an edge. In other words,
the cliques represent sites which are all neighbors to each other. Thus, a clique
consists of either a single site, or a pair of neighboring sites, or a triple, and so
on. The collection of single-site and pair-site cliques will be denoted by
S
C
1
and
C
2
respectively, where
C
1
=
{{
i
}|
i
∈
S
}
(3)
i, i
}|
i
∈
{{
N
i
,i
∈
S
}
.
C
2
=
(4)
The energy function
U
(
f
) is a measure of the likeliness of the occurrence of
f
for
a given model. For single-site and pair-site cliques, it is defined as
)=
i∈
S
V
1
(
f
i
)+
i∈
S
V
2
(
f
i
,f
i
U
(
f
)
(5)
i
∈
N
i
where
V
1
and
V
2
denote potential functions for single-site and pair-site cliques.
Lower energy of the joint distribution represents a better fit of the model to the
data.
When applied to digital images, the sites correspond to pixel locations, and
the neighborhood system is usually either 4-connectedness or 8-connectedness.
For a 4-connected system, the four types of pair-site clique that any non-edge
pixel belongs to are shown in figure 1.
A type of MRF of particular interest to labeling problems in computer vision
is the Multi-Level Logistic (MLL) model. In an MLL, the potential functions are
defined as
V
1
(
f
i
)=
α
f
i
(6)
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