Clustering of Medical Images - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 156

Intuitionistic fuzzy set is constructed using Yager's intuitionistic fuzzy
complement and is written as

{ }

IFS αα 1 /

Ax = , (), ( μ x 1 − μ ( )) | x xX ∈

λ A A

and the hesitation degree is

( )) /

αα 1

π () x =− −− 1 μ () ( x 1 μ x ( 7. 2 1)

A A A

In this intuitionistic fuzzy clustering algorithm, the criterion function con-
tains two terms: (a) the intuitionistic-type objective function as in conven-
tional FCM and (b) the IFE .
A second objective function is introduced which is the IFE that aims in
maximizing the good points in the class. The goal is to minimize the entropy
of the histogram of an image. The hesitation values of all the elements in
each cluster are added, and then the entropy of each class is calculated. It
tells the amount of fuzziness or uncertainty present in the cluster. The sec-
ond function is

c

∑ π

*

*

1 − π

J = e

k

2 k

k = 1

∑

n

*

where π = ( 1 / N ) π , k ∈ [1, N ]. π ik is the hesitation degree of the i ith ele-

k ik

i = 1

ment in cluster k .
So, the final criterion function that contains two terms is minimized and
is as follows:

n c c

∑ ∑ ∑

* (,) *

*

ik m

2 1 − π

J = u dx v + π e , with m = 2

k

i ik k

i 1 k 1 k 1

= = =

where
d ( x i , v k ) is the Euclidean distance measure (or any distance measure)
between v k (cluster centre) of each region and x i (points in the pattern)
u ik is the membership value of the i th data ( x i ) in the k th cluster
c is the number of clusters
n is the number of data points

The hesitation degree is calculated using Equation 7.21, and the intuitionistic
fuzzy membership values are obtained as follows:

* =+π

uu
ik

( 7. 2 2)

ik ik