Clustering of Medical Images - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 154

μ A , μ B and v A , v B are the membership and non-membership degrees, respec-
tively. The intuitionistic fuzzy intersection similarity between two sets A
and B is written as

s (, μμ ) + s vv ( , )

1 AB AB 1

SAB (,) = ( 7.19)

IFS

2

where

∑
∑

n

( )

min( μ x ), () μ x

Ai Bi

i = 1

if μμμ ∪≠ 0

AB

n

s (, μμ ) =

( )

max( μ x ), () μ x

1 AB

Ai Bi

i = 1

1 if μμ ∪= 0

AB

and for the non-membership degree,

∑
∑

n

( )

min( vxvx ), ()

Ai Bi

sv v (, ) =

i = 1

1 AB

n

( )

max( vxvx ), ()

Ai Bi

i = 1

where μ A = {μ A ( x )} and v A = { v A ( x )}.
The new objective criterion is reformulated as

n c n c

∑ ∑ ∑ ∑ 1

IFS m

J UVX (,:) = μ xv − , with 0 < μ < n , μ = 1

m i k ik ik

ik

IFS

i = 1 k = 1 i = 1 k =

− = −1 is the distance between the data vector x i and cluster cen-
tre v k . On minimizing J UVX

xv S
i k IFS

IFS

IFS (,: , we get as in FCM

( )

− 1 /( m − 1 )

xv −

i k IFS

u = , ∀< < u , 1 k c , 1 << iN

ik ik

∑

( )

− 1 /( m − 1 )

c

xv −

i j

IFS

j = 1

And the centroid is computed as

∑
∑

n

i m

ux

i

v =

i = 1

k

n

m

u

ik

i = 1

The distance function solely depends on the membership and non-
membership values after the computation of centroids and before the next
iterations.