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Finite Mixture Model
For every
and ,
is independent of the individual sites
and
called a mixing parameter. We take
as the model
parameter set with
Consider two configurations
and
.
From (1) and
(2), we can compute the joint probability
distribution of x and y dependent on the model
parameters (
is treated as a set of random variables),
namely

(3) 
We can compute the marginal distribution of Y_{i}=y, dependent on
the parameter set :
This is the socalled finite mixture (FM) model.
The FM model is the most frequently employed statistical model,
due to its simple mathematical form [21,29]. But
it is clear that the FM model only describes the data
statistically. No spatial information about the data is utilized.
In other words, the FM model is spatially independent and can
therefore be specified fully by the histogram of the data.
However, images with the same intensity distribution may have
totally different structural properties. For this reason, FM model
is not complete.
To overcome this drawback, certain spatial properties, or
constraints, have to be incorporated into the model. Here, spatial
properties means quantifiable structural characteristics of an
object, such as size and shape. Under certain intensity
distributions, we want the model to be ``adaptive'' to structural
information or spatially dependent in order to fit the actual
image better. This leads to the consideration of MRF theory and
our HMRF model.
Next: Markov Random Field Theory
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Yongyue Zhang
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