Clustering of Medical Images - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 149

Substituting Equation 7.13 in u 11 in Equation 7.11, we have

( )

− 1 /( m − 1 )

1 − Kx v (,)

11

u =

11

− 1 /( m − 1 )

∑

c

( )

1 Kx v (,)

−

1 j

j = 1

In general,

( )

− 1 /( m − 1 )

1 − Kx v (,)

i k

u = (7.14)

ik

∑

c

( )

− 1 /( m − 1 )

1 − Kx v (,)

i j

j = 1

To compute the cluster centre, the equation for the objective function is

⎛ ⎞

n c

⎛

2

⎞

n

⎛

c

⎞

−− xv

2 ∑ ∑ ∑

i k

m

⎜ ⎟ −

JUV (,) = u tanh ⎜ ⎟ λ ⎜ u − 1 ⎟ ( 7.1 5 )

m ik i ik

2

σ

⎝ ⎠

⎝ ⎠

⎝ ⎠

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

Expanding this equation and taking the first derivative with respect to v and
equating to zero, we obtain

n

⎛

2

⎞

−− xv xv

δ
δ

J
v

( − )

∑

i 1

m 2 i 1

= 2 u ⎜ 1 − tanh ⎟ = 0 as /tanh ) ddx ( x

i 1

2 2

σ σ

1

⎝ ⎠

i = 1

2

=− 1 tanh x ( or)

n

⎛

2

⎞ ⎛

2

⎞

−− xv −− xv x

∑

i 1 i 1

m

u ⎜ 1 + tanh ⎟ ⎜ 1 − tanh ⎟ ( −−= v 1 ) 0 ( or )

i 1 i

2 2

σ σ

⎝ ⎠ ⎝ ⎠

i = 1

n

⎛

2

⎞

−− xv Hxvxv

∑

i 1

m

u ⎜ 1 + tanh ⎟ (,)( −= ) 0 ( or )

i 1 i 1 i 1

2

σ

⎝ ⎠

i = 1

n

⎛

2

⎞

−− xv x

∑

i 1

m

uHxv (,) ⎜ 1 + tanh ⎟

i 1 i 1 i

2

σ

⎝ ⎠

i = 1

n

⎛

2

⎞

−−

i xv v

∑

1

i m

− uHxv (,) ⎜ 1 + tanh ⎟ = 0

1 i 1 1

2

σ

⎝ ⎠

i = 1