Clustering of Medical Images - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 147

7.5 Kernel Clustering Methods
1. Kannan et al. [11] suggested a robust FCM algorithm to segment medi-
cal images. They modified the objective function where they replaced the
Euclidean distance by a hypertangent kernel function. The objective func-
tion as in Equation 7.4 is

n c

∑ 1

2
ϕϕ

m

JUV (,) = u ( x ) − ( v )

m ik i k

i = 1 k =

For hypertangent kernel

⎛

2

⎞

−− xv

i k

Kx v (,) =− 1 tanh ⎜ ⎟

i k

2

σ

⎝ ⎠

where σ is a parameter adjusted by the users.
The objective function using Equation 7.7 reduces to

n c

2 ∑

m

JUV (,) = u ( (, )) 1 − Kx v

m ik i k

i = 1 k = 1

Using Lagrangian multiplier, the objective function can be written as

n c n

∑

c

⎛ ⎞

∑ ∑ ∑

m

JUV (,) = 2 u ( (, )) 1 − Kx v − λ u − 1 ( 7.10 )

⎜ ⎟ m ik i k i ik

⎝ ⎠

i = 1 k = 1 i = 1 k = 1

Expanding this equation,

⎡

n

⎤

∑

i m i m

JUV (,) = 2 u ( (,)) 1 − Kx v ++ − u ( 1 Kx v ( , )

⎢
⎢

⎥
⎥

m 1 i 1 i ic

⎣ ⎦

i = 1

n

∑

− λ i (( uu u +++− )) 1

i 1 i 2 ic

i = 1