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Homework#3 Color Image Segmentation by Clustering Algorithm

Yingru Tian

Color Image Segmentation by Clustering Algorithm

1. K-means Clustering (HCM)

K-means Clustering is based on classification , the idea of which is to make sure the similarity

between ones from the same family is as large as possible, while that between ones from different

family is as small as possible . Its core idea is: classify n vectors

into c groups

To

function representing th e membership between the feature vectors and the clusterings. The

classified groups are generally defined using a

membership matrix U. If the

vector

belongs to group

, the element

in U is set to 1; if not, it's set to 0. Once the clustering center

is decided, we can have the following equation:

group

:

)2 .(

1

i

i

=

iji

Since HCM applies a rigid membership matrix (the elements of the membership matrix are

either 0 or 1), the classification results of this method are not perfect. Thus I will apply Fuzzy

C-mean method (FCM) to classified the original images.

2. Fuzzy C-mean method ( FCM )

2.1 Theory background

FCM has been improved compared with common C-mean method (HCM) . For HCM , it

applies rigid classification while FCM is not. FCM is a clustering algorithm which determines the

possibility when one point belong to a specified class, applying the degree of membership.

The membership function in FCM, which is a function representing the degree how sample x

set A can be expressed as:

})),({(

XxxxA

tttA

∈=

µ

According to the concept of fuzzy set , every degree of the sample points belong to its

Homework#3 Color Image Segmentation by Clustering Algorithm

Yingru Tian

family is the value from [0,1]

2.2 FCM

FCM divides n vector s into c fuzzy sets , and calculate the the clustering center of each fuzzy

set, mak ing sure the dissimilarity index as a cost function is minimum. In this report, I use

Euclidean distance and membership function to define the cost function.

w here

is the element of membership matrix U at position (i,j).

is Euclidean distance

between i

clustering center and sample

By the normalization rules, we can get a data set, of which the total degree of membership

equal to 1

minimizing the value of (Eq 3 ):

)5.()1(

)1(),...,,(),...,,,...,,(

uccUJccUJ

n

j

c

i

j

i

∑ ∑∑ ∑

−+=

−+=

λ

)6.(

d

d

and

Eq

u

xu

c

k

m

j

ij

i

∑

⎜

⎜

=

Step

Step 1:

1: Apply the function rand () to produce a random initialized membership matrix U with

Step

Step

2:

2:

Step

Step

Step

Step 3:

3:

3:

3: According to (Eq . 3 ), we calculate the cost function. If it is less than a defined

Homework#3 Color Image Segmentation by Clustering Algorithm

Yingru Tian

threshold, or its value relative to the last change of function value is less than a threshold, then the

algorithm stops.

Step

Step

Step

Step 4:

4:

4:

4: Calculate the new membership matrix U using (Eq . 7). Return to Step 2.

The outputs of the algorithm are a clustering center vector and a fuzzy partition

matrix(

), which represents the degree of membership of each sample belonging to every