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.
 
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