Image Processing Reference

In-Depth Information

mentation of brain MR images along with the initialization method of c-means algorithm

based on the maximization of class separability. Implementation details, experimental re-

sults, and a comparison among different c-means are presented in Section 2.6. Concluding

remarks are given in Section 2.7.

2.2

Fuzzy C-Means and Rough Sets

This section presents the basic notions of fuzzy c-means and rough sets. The rough-fuzzy

c-means (RFCM) algorithm is developed based on these algorithms.

2.2.1 Fuzzy C-Means

Let X ={x

,, v
i
,, v
c
}be the

set of c centroids, where x
j
∈ℜ
m
, v
i
∈ℜ
m
, and v
i
∈X. The fuzzy c-means provides a

fuzzification of the hard c-means (Bezdek, 1981; Dunn, 1974). It partitions X into c clusters

by minimizing the objective function

,, x
j
,, x
n
}be the set of n objects and V ={v

1

1

n

c

(
ij
)
´m
||x
j
−v
i
||
2

J =

(2.1)

j=1

i=1

where 1≤m <∞is the fuzzification factor, v
i
is the ith centroid corresponding to cluster

β
i
,
ij
∈[0, 1] is the fuzzy membership of the pattern x
j
to cluster β
i
, and||.||is the distance

norm, such that

n

n

1

n
i

(
ij
)
´m
x
j
; where

(
ij
)
´m

v
i
=

n
i
=

(2.2)

j

=1

j

=1

and

c

(
d
ij

d
kj

2

m−1
)
−1
; where

d
2
ij

=||x
j
−v
i
||
2

ij
= (

)

(2.3)

k=1

subject to

c

n

ij
= 1,∀j, and 0 <

ij
< n,∀i.

i=1

j=1

The process begins by randomly choosing c objects as the centroids (means) of the c

clusters. The memberships are calculated based on the relative distance of the object x
j
to

the centroids by Equation 2.3. After computing memberships of all the objects, the new

centroids of the clusters are calculated as per Equation 2.2. The process stops when the

centroids stabilize. That is, the centroids from the previous iteration are identical to those

generated in the current iteration. The basic steps are outlined as follows:

1. Assign initial means v
i
, i = 1, 2,, c. Choose values for m and threshold ǫ. Set

iteration counter t = 1.

2. Compute memberships
ij
by Equation 2.3 for c clusters and n objects.

3. Update mean (centroid) v
i
by Equation 2.2.

4. Repeat steps 2 to 4, by incrementing t, until|
ij
(t)−
ij
(t−1)|> ǫ.

Although fuzzy c-means is a very useful clustering method, the resulting memberships

values do not always correspond well to the degrees of belonging of the data, and it may

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