Biomedical Engineering Reference
In-Depth Information
Finally, to show the convergence of the proposed model, we will show ex-
perimentally, when we use this model, the Levy distance will decrease between
the estimated distribution
P
es
(
y
) and empirical distribution
P
em
(
y
). The Levy
distance
ρ
(
P
em
,
P
es
) is defined as
ρ
(
P
em
,
P
es
)
=
inf
{
ξ>
0:
∀
yP
em
(
y
−
ξ
)
−
ξ
≤
P
es
(
y
)
≤
P
em
(
y
+
ξ
)
+
ξ
}
.
(9.22)
As
ρ
(
P
em
,
P
es
) approach zero,
P
es
(
y
) converge weakly to
P
em
(
y
).
9.2.6 Parameter Estimation for High-Level Process
In order to carry out the MAP parameters estimation in Eq. 9.5, one needs to
specify the parameters of high-level process. A popular model for the high-level
process is the Gibbs Markov model which follows Eq. 9.2. In order to estimate
the parameters of GMRF, we will find the parameters that maximize Eq. 9.2, and
we will use the Metropolis algorithm and genetic algorithm (GA).
The Metropolis algorithm is a relaxation algorithm to find a global maximum.
The algorithm assumes that the classes of all neighbors of
y
are known. The high-
level process is assumed to be formed of
m
-independent processes; each of the
m
processes is modeled by Gibbs Markov random which follow Eq. 9.2. Then
y
can be classified using the fact that
p
(
x
i
|
y
) is proportional to
p
(
y
|
x
t
)
P
(
x
t
|
η
s
),
where
s
is the neighbor set to site
S
belonging to class
x
t
,
p
(
x
t
|
η
s
) is computed
from Eq. 9.2, and
p
(
y
|
x
t
) is computed from the estimated density for each class.
By using the Bayes classifier, we get initial labeling image. In order to run the
Metropolis algorithm, first we must know the coefficients of potential function
E
(
x
), so we will use GA to estimate the coefficient of
E
(
x
) and evaluate these
coefficients through the fitness function.
9.2.6.1 Maximization Using Genetic Algorithm
To build the genetic algorithm, we define the following parameters:
Chromosome:
A chromosome is represented in binary digits and consists of
representations for model order and clique coefficients. Each chromosome has
41 bits. The first bit represent the order of the system (we use digit “0” for first-
order and digit “1” for second-order-GMRF). The remaining bits represent the
