Image Processing Reference

In-Depth Information

if x
j
∈B(β
i
)⇒∃k, x
j
∈B(β
k
). It means an object x
j
∈B(β
i
) possibly belongs to β
i

and potentially belongs to other cluster. The properties 5 and 6 are of great importance

in computing the objective function J
RF
and the cluster prototype v
RF
. They say that

the membership values of the objects in lower approximation are
ij
= 1, while those in

boundary region are the same as fuzzy c-means. That is, each cluster β
i
consists of a crisp

lower approximation A(β
i
) and a fuzzy boundary B(β
i
).

2.3.3

Details of the Algorithm

Approximate optimization of J

(Equation 2.4) by the RFCM is based on Picard iteration

through Equations 2.3 and 2.5. This type of iteration is called alternating optimization.

The process starts by randomly choosing c objects as the centroids of the c clusters. The

fuzzy memberships of all objects are calculated using Equation 2.3.

Let
i
= (
i1
,,
ij
,,
in
) represent the fuzzy cluster β
i
associated with the centroid

v
i
. After computing
ij
for c clusters and n objects, the values of
ij
for each object

x
j
are sorted and the difference of two highest memberships of x
j
is compared with a

threshold value δ. Let
ij
and
kj
be the highest and second highest memberships of x
j
.

If (
ij
−
kj
) > δ, then x
j
∈A(β
i
) as well as x
j
∈A(β
i
), otherwise x
j
∈A(β
i
) and

x
j
∈A(β
k
). After assigning each object in lower approximations or boundary regions of

different clusters based on δ, memberships
ij
of the objects are modified. The values of

ij
are set to 1 for the objects in lower approximations, while those in boundary regions are

remain unchanged. The new centroids of the clusters are calculated as per Equation 2.5.

The main steps of the RFCM algorithm proceed as follows:

RF

1. Assign initial centroids v
i
, i = 1, 2,, c. Choose values for fuzzification factor

m, and thresholds ǫ and δ. Set iteration counter t = 1.

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

3. If
ij
and
kj
be the two highest memberships of x
j
and (
ij
−
kj
)≤δ, then

x
j
∈A(β
i
) and x
j
∈A(β
k
). Furthermore, x
j
is not part of any lower bound.

4. Otherwise, x
j
∈A(β
i
). In addition, by properties of rough sets, x
j
∈A(β
i
).

5. Modify
ij
considering lower and boundary regions for c clusters and n objects.

6. Compute new centroid as per Equation 2.5.

7. Repeat steps 2 to 7, by incrementing t, until|
ij
(t)−
ij
(t−1)|> ǫ.

The performance of the RFCM depends on the value of δ, which determines the class

labels of all the objects. In other word, the RFCM partitions the data set into two classes

- lower approximation and boundary, based on the value of δ.

In the present work, the

following definition is used:

n

1

n

δ =

(
ij
−
kj
)

(2.6)

j=1

where n is the total number of objects,
ij
and
kj
are the highest and second highest

memberships of x
j
. That is, the value of δ represents the average difference of two highest

memberships of all the objects in the data set. A good clustering procedure should make

the value of δ as high as possible. The value of δ is, therefore, data dependent.

2.4

Pixel Classification of Brain MR Images

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