Biomedical Engineering Reference
In-Depth Information
component. To estimate the distribution for each class, we use the expecta-
tion maximization algorithm. The first step to estimate the distribution for each
class is to estimate the dominant Gaussian components in the given empirical
distribution.
9.2.5.1 Dominant Gaussian Components Extracting Algorithm
1. Assume the number of Gaussian components that represent the classes
i
,
i
=
1
, ...,
m
. Initialize the parameters of each distribution randomly.
2.
The E-step
: Compute
δ
it
that represent responsibility that the given pixel
value is extracted from certain distribution as
i
p
(
y
t
|
i
,
i
)
π
k
it
=
t
=
1to
N
2
δ
l
=
1
π
l
,
l
)
,
for
,
(9.8)
k
k
l
p
(
y
t
|
i
is the mix-
where
y
t
is the gray level at location
t
in the given image,
π
k
ing proportion of Gaussian component
i
at step
k
, and
i
is estimated
parameter for Gaussian component
i
at step
k
.
3.
The M-step
: we compute the new mean, the new variance, and the new
proportion from the following equations:
t
=
1
δ
it
,
N
2
k
+
1
i
(9.9)
π
=
N
2
t
=
1
it
y
t
δ
k
+
1
i
µ
=
,
(9.10)
N
2
t
=
1
δ
k
it
N
2
t
=
1
δ
it
(
y
t
−
µ
i
)
2
k
+
1
i
)
2
(
σ
=
.
(9.11)
N
2
t
=
1
k
it
δ
4. Repeat steps 1 and 2 until the relative difference of the subsequent values
of Eqs. 9.9, 9.10, and 9.11 are sufficiently small.
Let
p
I
1
(
y
),
p
I
2
(
y
),...,
p
I
m
(
y
) be the dominant Gaussian components that
are estimated from the above algorithm. Then the initial estimated density
(
p
I
(
y
)) for the given image can be defined as follows:
p
I
(
y
)
=
π
1
p
I
1
(
y
)
+
π
2
p
I
2
(
y
)
+···+
π
m
p
I
m
(
y
)
.
(9.12)
Because the empirical data does not exactly follow mixture of normal distri-
bution, there will be error between
p
I
(
y
) and
p
em
(
y
). So we suggest the following
