Biomedical Engineering Reference
In-Depth Information
9.4.4.5 Cluster Prototype Updating
Taking the derivative of
F
m
w.r.t.
v
i
and setting the result to zero, we have
N
N
u
ik
N
R
u
ik
(
y
k
−
β
k
−
v
i
)
+
(
y
r
−
β
r
−
v
i
)
v
i
=
v
i
=
0
.
(9.41)
k
=
1
k
=
1
y
r
∈
N
k
Solving for
v
i
, we have
k
=
1
u
ik
(
y
k
−
β
k
)
+
N
R
y
r
∈
N
k
(
y
r
−
β
r
)
(1
+
α
)
k
=
1
u
ik
v
i
=
(9.42)
.
9.4.4.6 Bias Field Estimation
In a similar fashion, taking the derivative of
F
m
w.r.t.
β
k
and setting the result to
zero we have
u
ik
(
y
k
−
β
k
−
v
i
)
2
c
N
∂
∂β
k
β
k
=
β
k
=
0
.
(9.43)
i
=
1
k
=
1
Since only the
k
th term in the second summation depends on
β
k
, we have
∂β
k
u
ik
(
y
k
−
β
k
−
v
i
)
2
c
∂
β
k
=
β
k
=
0
.
(9.44)
i
=
1
Differentiating the distance expression, we obtain
c
c
c
u
ik
−
β
k
u
ik
−
u
ik
v
i
y
k
β
k
=
β
k
=
0
.
(9.45)
i
=
1
i
=
1
i
=
1
Thus, the zero-gradient condition for the bias field estimator is expressed as
i
=
1
u
ik
v
i
β
k
=
y
k
−
i
=
1
u
ik
.
(9.46)
9.4.4.7 BCFCM Algorithm
The BCFCM algorithm for correcting the bias field and segmenting the image
into different clusters can be summarized in the following steps:
i
=
1
. Set
{
β
k
}
k
=
1
to equal and very
Step 1.
Select initial class prototypes
{
v
i
}
small values (e.g. 0.01).
Step 2.
Update the partition matrix using Eq. 9.40.