Biomedical Engineering Reference
In-Depth Information
mass function in the following form
1
Z e
E ( X )
T
p ( x ) =
,
(9.2)
where Z = x e E ( x )
, and the function E ( x ) is called energy function.
T
9.2.3 Markov Random Fields
Hassner and Sklansky introduced Markov random fields to image analysis and
throughout the last decade Markov random fields have been used extensively
as representations of visual phenomena. A Gibbs random filed describes the
global properties of an image in terms of the joint distributions of colors for all
pixels. An MRF is defined in terms of local properties. Before we show the basic
properties of MRF, we will show some definitions related to Gibbs and Markov
random fields [10-15].
Definition 1 : A clique is a subset of S for which every pair of sites is a
neighbor. Single pixels are also considered cliques. The set of all cliques on a
grid is called .
Definition 2 : A random field X is an MRF with respect to the neighborhood
system η ={ η s , s S } if and only if
p ( X = x ) > 0 for all x , where is the set of all possible configurations
on the given grid;
p ( X s = x s | X s | r = x s | r ) = p ( X s = x s | X s = x s ), where s | r refers to all N 2
sites excluding site r , and s refer to the neighborhood of site s ;
p ( X s = x s | X s = x s ) is the same for all sites s .
The structure of the neighborhood system determines the order of the MRF.
For a first-order MRF the neighborhood of a pixel consists of its four nearest
neighbors. In a second-order MRF the neighborhood consists of the eight nearest
neighbors. The cliques structure are illustrated in Figs 9.1 and 9.2.
Consider a graph ( t ) as shown in Fig. 9.3 having a set of N 2 sites. The
energy function for a pairwise interaction model can be written in the following
form:
N 2
N 2
w
E ( x ) =
F ( x t ) +
H ( x t , x t : + r ) ,
(9.3)
t = 1
t = 1
r = 1
 
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