Image Processing Reference
In-Depth Information
∑
n
(
)
m
ux x
+
α
ik
i
i
v
=
i
=
1
,
i
= …= …
123
, ,, ,,
n
k
1 23
,,,
,
c
k
∑
n
m
(
1
+
α
)
u
ik
i
=
1
When α
= 0, the modified algorithm converges to the original FCM.
x
i
is
taken as the median of the neighbours in a window around
x
i
.
The Euclidean distance ∥
x
i
−
v
k
∥ is replaced with Gaussian kernel distance
1 −
K
(
x
i
,
v
k
) = 1 − exp(−∥
x
i
−
v
k
∥/σ
2
).
So, the new objective function with the kernel version is written as
c
n
c
n
∑
∑
∑
∑
(
)
+
(
)
m
m
JUV
(,)
=
u
1
−
Kx v
(
, )
α
u
1
−
Kx v
(,
k
)
( 7.18)
m
ik
i k
ik
i
k
=
1
i
=
1
k
=
1
i
=
1
The necessary conditions to minimize are as follows:
−
1
/(
m
−
1
)
(
)
1
−
Kx v
(,)
+
α
( (, ))
1
−
K xv
i k
i k
u
=
ik
−
1
/(
m
−
1
)
∑
1
c
(
)
(
1
−
Kx v
( , )
+
α
(
1
−
K x
(
, )
i
j
i
j
j
=
∑
μ
n
i
m
Kxvx Kxvx
(,)
+
α
(
, )
i ki
i ki
v
=
i
1
k
∑
1
n
m
μ
((, ) (,
Kxvx Kx v
+
α
))
ik
i ki
i
k
i
The usual clustering procedure then follows.
E x a mple 7. 2
An example is shown on CT scan images of the brain about the efficacy
of the kernel function in the original FCM algorithm along with spatial
information (Figure 7.4).
It is observed that fuzzy clustering with kernel can detect the blood
clot clearly as the kernel function transforms the input image to a high-
dimensional feature space. Now, the effect of intuitionistic fuzzy set on
medical image clustering is shown where more uncertainties are consid-
ered. It may be useful in those images where the fuzzy method may not give
better results.
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