Clustering of Medical Images - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 159

where from Equation 7.6

2

ϕϕ () () (,) x − v = KxxKvv Kxvi + ( , ) (,) − 2 = 1 2 , , … , n and k = 1 2 ,, …, c

i k i i kk i ik

where
i is the data point
k is the cluster centre

Three kernels, namely, the hypertangent kernel, Gaussian kernel and radial
basis kernel are used.
For all these kernels, K ( x i , x i ) = 1 and K ( v k , v k ) = 1.
The objective function in FCM reduces to

n c

2 ∑

m

JUV (,) = u ( (, )) 1 − Kx v

m ik i k

i = 1 k = 1

The membership matrix is

( )

− 1 /( m − 1 )

1 − Kx v (,)

i k

u =

ik

∑

c

( )

− 1 /( m − 1 )

1 − Kx v (,)

i j

j = 1

*

The modified intuitionistic fuzzy membership degree is uu
ik

=+π u ik
is the original membership matrix in FCM where π ik = 1 − u ik − (1 − u ik )/
(1 + λ ⋅ u ik ), and λ is taken as 1. Sugeno's fuzzy complement is used to compute
the non-membership degree.
The cluster centre after incorporating the intuitionistic property using
hypertangent kernel is written as

;

ik ik

⎛ ⎞

⎛

2

⎞

−− xv x

∑ μ

n

* (,)

k i

m

⎜
⎜

⎟
⎟

Kx v 1 + tanh ⎜ ⎟

ik i k i

2

σ

k 1

⎝ ⎠

⎝ ⎠

v =

i

⎛ ⎞

⎛

2

⎞

−− xv

∑

n

*

k i

m

⎜ ⎟

μ Kx v (,) 1 + tanh ⎜ ⎟

ik i k

2

σ

k = 1

⎝ ⎠

⎝ ⎠

The cluster centre using Gaussian and radial basis kernel is written as

∑
∑

n

* (,)

m

μ Kxvx

ik i ki

v =

k = 1

( 7. 2 4)

i

n

* (,)

m

μ Kx v

ik i k

k = 1