Clustering of Medical Images - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 142

between the hard and fuzzy c partitions of data set X . It is an iterative algo-
rithm where the aim is to find the cluster centres that minimize the dissimi-
larity function. This is an important feature in medical image diagnosis to
increase the sensitivity.

7.2 Fuzzy c Means Clustering
The problem of fuzzy clustering may be viewed as an extension of crisp clus-
tering. The algorithm requires a priori definition of a number of classes that
will partition the image in different classes. It classifies the set of data points
X = ( x 1 , x 2 , x 3 , …, x n ) into c homogeneous groups or clusters represented as
fuzzy sets, F = ( F 1 , F 2 , F 3 , …, F c ).
Let X = ( x 1 , x 2 , …, x j , …, x n ) be a set of sample, where data point x k ( k = 1,
2, …, n ) is required to be partitioned in c (2 ≤ c ≤ n ) clusters. Let us assume
that u ik is the membership grade of pattern x k to the cluster i and U = [ u ik ] is a
c × n membership matrix where u ik is the membership grade of the i th object
in the k th group:

c
c

uu u
u

⎡ ⎤

n

1 11 12 1

⎢
⎢
⎢
⎢

⎥
⎥
⎥
⎥

2 21

U =

c u u

⎣ ⎦

n c cn

1

The membership matrix implies that the n th data, x n , belong to class c 1 ,
c 2 , …, c c , with membership functions u 1 n , u 2 n , …, u cn , respectively. The mem-
bership distribution has the following properties:

u ik ∈ [0, 1], ∀ i , k i = 1, 2, 3, …, n , k = no. of classes

n

i

<

ik un kc

< ≤≤

0 1

= 1

and

c

∑ =

u 10 1 . ≤ ≤ j n

ik

k = 1