Brain Tumor Detection Using Image Processing Techniques

133

points and the cluster centroids are computed using Equation 4.34 [51]:

J =

k



j=1

n



i=1

x(j)

i

−c(j)



2

(4.34)

In the equation k stands for the number of clusters, n stands for a number

of data points and cj stands for the centroid for cluster j. The distance of

each data point (xi) to the centroid of each cluster (cj) is calculated. Data

point is assigned to the nearest cluster. After the placement of all data points

is completed, the new cluster centroids are computed for each cluster. Data

points are reassigned according to these new cluster centroids. These steps are

repeated until the centroids of the clusters are fixed or until the number of

iterations is reached. Thus, the image is segmented into regions, in our case

the brain is divided into parts.

Fuzzy C-means Clustering Algorithm

In fuzzy clustering, unlike classical clustering, data may belong to more than

one cluster, in other words, there are partial memberships of the data. In

this method, membership degrees between [0,1] are assigned to the data to

indicate the degree to which the data belongs to different clusters. In this

case, the membership degree will be the highest for the cluster nearest to the

data. This algorithm is designed to minimize the objective function given in

Equation 4.35 [51]:

Jm =

N



i=1

c



j=1

u^m

ij ^{∥xi −vj∥2 , 1 ≤m < ∞}

(4.35)

In the equation, N represents the total number of data points, c denotes

the number of clusters, uij signifies the membership degree of data point xi

in cluster j, xi stands for the ith data point, vj indicates the center of cluster

j, and m is the exponent of the partition matrix [51].

The algorithm first assigns a number of clusters and randomly initializes

the cluster centers. In the next step, the distances between the data points

and the center vectors are calculated and membership degrees are updated

using Equation 4.36 [52]:

uij =

1

c

k=1



∥xi−vj∥

∥xi−vk∥



2

m−1

(4.36)

Then cluster centers are computed using Equation 4.37 [52]:

vj =

N

i=1 ^um

ij^.xi

N

i=1 ^um

ij

(4.37)