Biomedical Engineering Reference
In-Depth Information
and
N
0
<
u
ik
<
N
∀
i
}
.
(9.24)
k
=
1
The parameter
p
is a weighting exponent on each fuzzy membership and
determines the amount of fuzziness of the resulting classification. The FCM
objective function is minimized when high membership values are assigned to
pixels whose intensities are close to the centroid of its particular class, and low
membership values are assigned when the pixel data is far from the centroid.
9.4.2 Modified Fuzzy C-Means Objective Function
We propose a modification to Eq. 9.23 by introducing a term that allows the
labeling of a pixel to be influenced by the labels in its immediate neighborhood.
As mentioned before, the neighborhood effect acts as a regularizer and biases
the solution toward piecewise-homogeneous labeling. Such a regularization is
useful in segmenting scans corrupted by salt and pepper noise. The modified
objective function is given by
c
N
u
ik
||
x
k
−
v
i
||
2
J
m
=
i
=
1
k
=
1
(9.25)
x
r
∈
N
k
||
x
r
−
v
i
||
2
c
N
+
N
R
u
ik
,
i
=
1
k
=
1
where
N
k
stands for the set of neighbors that exist in a window around
x
k
and
N
R
is the cardinality of
N
k
. The effect of the neighbors term is controlled by
the parameter
α
. The relative importance of the regularizing term is inversely
proportional to the signal to noise ratio (SNR) of the image signal. Lower SNR
would require a higher value of the parameter
α
.
Formally, the optimization problem comes in the form
min
U
,
{
v
i
}
J
m
subject to
U
∈
U.
(9.26)
i
=
1
9.4.3 Parameter Estimation
The objective function
J
m
can be minimized in a fashion similar to the standard
FCM algorithm. Taking the first derivatives of
J
m
with respect to
u
ik
and
v
i
, and
