Clustering of Medical Images - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 143

The objective is to obtain a c partition by minimizing the criterion function
using the Lagrangian multiplier method:

n c

∑ ∑

m 2

JUVX (,:) = μ dxv (,) ( 7.1)

m ik i )ik

i = 1 k = 1

The membership matrix U is randomly initialized as

1

u = , ∀= …= k 12 , , , c , i 12 , , … , n ( 7. 2)

ik

( )

∑

c

( )

2 /( m − 1 )

( )

xv xv − −

i k i j

j = 1

and the cluster centres are computed as

∑
∑

n

m

ux

ik i

v =

i = 1

( 7. 3)

k

n

m

u

ik

i = 1

where
u ik is the membership of the data x i to the k th fuzzy cluster with centroid v k
m is user defined, and generally it is taken as 2
' d ' is the Euclidean distance, or any distance measure is used to find the
similarity between the cluster centre and the data points

This iteration will stop when max{| ε , where ε is a tolerance

level which lies in between 0 and 1. t and t + 1 are the successive iterations.
An example of fuzzy clustering is shown in Figure 7.1.
Many studies are reported in the literature to improve the fuzzy clustering
algorith m [10].

u

( t + 1 )

− u

() t

|} <

ik

ik ik

(a) (b)

FIGURE 7.1
(a) Brain image and (b) fuzzy clustering into four regions.