Image Processing Reference
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 ).
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:
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:
( ij − kj )
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.
Pixel Classification of Brain MR Images