Clustering of Medical Images - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 150

2

⎛ ⎞

−− xv x

∑

n

i 1

m

uHxv (,) ⎜ 1 + tanh ⎟

i 1 i 1 i

2

σ

i = 1

⎝ ⎠

v = ( 7.16 )

1

2

⎛ ⎞

−−

i xv

∑

n

1

m

uHxv (,) ⎜ 1 + tanh ⎟

i 1 i 1

2

σ

i = 1

⎝ ⎠

Likewise, for v k ,

2

⎛ ⎞

−− xv x

∑

n

i k

m

uHxv (,) ⎜ 1 + tanh ⎟

ik i k i

2

σ

i = 1

⎝ ⎠

v =

k

2

⎛ ⎞

−− xv

∑

n

i k

m

uHxv (,) ⎜ 1 + tanh ⎟

ik i k

2

σ

i = 1

⎝ ⎠

Following a similar procedure as FCM, iteration will stop when
{|

( t + 1 )

() t

ε where ε is a tolerance level which lies in between 0 and 1.
t and t + 1 are successive iterations.

u − u |} < ,

ik

ik

E x a mple 7.1

Two examples on computed tomography (CT) scan images of the brain
are shown to show the effectiveness of the kernel function in the original
FCM algorithm (Figure 7.3).

2. To increase the robustness of FCM, Ahmed et al. [1] considered spatial
neighbourhood information and the objective function is modified as

n c n

∑

c

α

∑ ∑ ∑ ∑

2 2

m m

JUV (,) = u xv N − + u x − v

m i k ik r k

iikk

R

i = 1 k = 1 i = 1 k = 1 xN ∈

r i

where
N i is the set of pixel neighbours in a window around x i
N R is the cardinality of N i
α is the controlling parameter of the neighbouring terms
x i is the sample mean within the window in the neighbourhood of x i

The membership and cluster centres are updated as

( )

− 1 /( m − 1 )

∑

2 2

xv N −+ ( α / ) x − v

i k R r k

xN ∈

u =

r i

ik

∑ ( ) 1

− /( m − 1 )

c

∑

2 2

xv N −+ ( α / ) x − v

i j R r k

j = 1 xxN ∈

r i