Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 3, No. 3, 2022 

 

70 

Research	on	Clustering	Methods	for	Color	Image	
Segmentation	
Qingzhen Gong1, 2, * 

1 School of Physics and Electronic Information Engineering, Jining Normal University, Ulanqab City, Inner Mongolia, 012000, China 
2 Center for International Education, Philippine Christian University, Manila City, 1004 Metro Manila, Philippines 
* Corresponding Email: xunuo1314_2012@163.com 

 

Abstract: The research of image segmentation mainly includes: how to select the appropriate color space, reduce the 
complexity of segmentation algorithm, improve the noise resistance and universality of segmentation algorithm, etc. Fuzzy 
clustering is an unsupervised classification method, which can classify samples with similar properties without prior knowledge. 
This paper briefly describes the working principle and performance comparison of these algorithms. 

Keywords: Color image segmentation, Fuzzy clustering, Fuzzy C-Means, K-Means. 
 

1. Introduction 
On clustering and segmentation is a basic step of image 

processing. The purpose of this operation is to separate 
different homogeneous areas in the image, so as to organize 
objects into groups (clusters) whose members have different 
common attributes (intensity, color, texture, etc.). 
Segmentation methods can be divided into two categories: 
Fuzzy C-Means and K-Means. Firstly, the algorithms of these 
two methods and their characteristics are introduced. 
Secondly, the difficulties and defects of Fuzzy C-Means 
algorithm in image segmentation are analyzed. Finally, we 
will implement these two algorithms through a series of 
examples of testing functions realized by MATLAB. 

2. Algorithm Description 
Fuzzy C-Means is the most commonly used image 

segmentation method based on fuzzy theory and cluster 
analysis. It is a clustering method based on the idea of 
partition, which makes the samples belonging to the same 
area have the greatest similarity through partition, while the 
samples divided into different areas are the ones with the 
smallest similarity. 

2.1. K-Means 
Means algorithm is Fuzzy C-Means. The main idea of this 

algorithm is to divide N samples xj(1,2...,n) into C groups 
Gi(i=1,2...,c), so as to minimize the objective function and 
obtain the cluster center ci of each group. Usually, the function 
d(xk,ci) is used to represent the dissimilarity between samples, 
and the objective function is defined as: 

 

 
 


c

i xk

c

i k

ik cxdJiJ
1 G,1

))((
i

       (1) 

 

When the value of the function d(xk,ci) is the Euclidean 
distance between the sample xk in group I and the cluster 
center ci of the category to which it belongs, the objective 
function can be expressed as: 

 
 


c

i xk

c

i k

ik cxJiJ
1 G,

2

1 i

)(           (2) 

 
If ci is the nearest cluster center to the sample xj, then xj 

belongs to the i-th group. The divided group can be 
represented by a c×n two-dimensional membership matrix U. 
When the sample xj belongs to the i-th group, the element uij 
in U is 1, otherwise it takes 0, which can make the objective 
function J reach the minimum value. If the cluster center ci is 
determined, U is determined according to the following 
formula: 

 






0

1
jiu

else

cxcxik kjij
22

, 
     (3) 

 
Since each sample can only belong to one group, uij should 

satisfy the following two formulas: 

 

,1
1




c

i

iju nj ,...,1              (4) 

 

nu
c

i

n

j

ij 
 1 1

               (5) 

 
If the membership matrix U is determined, then the cluster 

center ci of each group that minimizes the objective function 
J is the average value of the samples in the i-th group 
expressed by the following formula. 

 





ik

k

i

i

Gxk
x

G
c

,

1
              (6) 

 
Where |Gi| is the number of samples in Group I of the 

following formula. 





n

j

iji uG
1

               (7) 

 



 

71 

Means algorithm is to repeatedly iterate the above process, 
namely: 

Step1: Selecting c values from a sample space as initial 
clustering centers; 

Step 2: Calculate the membership matrix U by Formula 2-
3; 

Step 3: Find the objective function J from formula 2-2, and 
when its change is less than a certain threshold, the iteration 
ends; 

Step 4: Get a new cluster center from Formula 2-6, and go 
to Step 2. 

2.2. Fuzzy C-Means 
Fuzzy C-Means algorithm divides n samples xj(1,2...,n) 

into c fuzzy groups Gi(i=1,2,...,c) to minimize the objective 
function and obtain the cluster center ci of each group. The 
difference between it and hard C partition is that the Fuzzy C-
Means algorithm uses the value in the interval [0,1] to 
indicate the degree to which each sample belongs to a certain 
group, but the sum of a sample's membership degree to each 
group must be 1, as follows: 

 

njij

c

i

u ,...,1,1

1




              (8) 

 
The objective function is changed from Formula 2-2 to the 

following formula: 

 


 


c

i

n

j
ij

m

ij

c

i

i duJccUJ
1 1

2

1

21 ),...,,(         (9) 

 
Where dij=||ci-xj|| is the Euclidean distance between the j-th 

sample xj and the i-th cluster center ci; The value of uij is in 
the interval [0,1]; The weighted index m=2 is usually taken. 

Find each cluster center ci and each element uij in the 
membership matrix U with the minimum value of the 
objective function J through the functions in the following 
construction formula. 

 
 


n

j

c

i

ijjcnc uccUJccUJ
1 1

111 )1(),...,,(),...,,,...,,(   

 
 


n

j

c

i

ijjij

c

i

n

j

m

ij udu
1 1

2

1

)1(            (10) 

 
Where j (j=1...n) is n Lagrangian factors, and we find the 

conditions for varying the input parameters to obtain the 
minimum value of J, that is: 







 n

j

m

ij

n

j

j
m

ij

i

u

xu
c

1

1
               (11) 












c

k

m

kj

ij

ij

d
d

u

1

)1/(2

1

            (12) 

 
Fuzzy C-Means algorithm steps are similar to K-Means. 
Step 1: Initializing the membership matrix U, and taking 

any value in the interval [0,1] for the elements in U, while 
meeting the requirements of 2-8; 

Step 2: Calculating the CI of each cluster center from 2 to 
11; 

Step 3: Find the objective function J from 2-9, and when its 
change is less than a certain threshold, the iteration ends; 

Step 4: Get a new membership matrix U from 2-12, and go 
to step 2. 

3. Algorithm Characteristics 
K-Means 
Advantages: K-Means algorithm is a classical algorithm to 

solve clustering problems, which is simple and fast; For large 
data sets, the algorithm is relatively scalable and efficient, 
because its complexity is about O(nkt), where n is the number 
of all objects, k is the number of clusters, and t is the number 
of iterations. Generally k<n . This algorithm often ends in 
local optimum; The algorithm tries to find out k partitions that 
minimize the square error function value. When clusters are 
dense, spherical or clustered, and the difference between 
clusters is obvious, its clustering effect is very good. 

Disadvantages: it is necessary to define the mean value; 
Specify the number to cluster; Some excessive outliers will 
have great influence; The algorithm is sensitive to initial value 
selection; Suitable for spherical clustering. 

Fuzzy C-Means 

Advantages: Fuzzy C-Means functional Jm is still a natural 
extension of traditional hard C-Means functional J1. J1 is a 
widely used clustering criterion, and its theoretical research 
has been quite perfect, which provides good conditions for the 
research of Jm. This algorithm has not only been successfully 
applied in many fields, but also based on this algorithm, fuzzy 
clustering algorithms based on other prototypes have been put 
forward, forming a number of Fuzzy C-Means algorithms, 
such as Fuzzy C-Line(FCL), fuzzy C-plane (FCP), fuzzy C-
shell (FCS), etc., which respectively realize linear, hyperplane 
and "thin shell" clustering. 

Disadvantages: Although fuzzy clustering is an 
unsupervised classification, FCM-type algorithm needs to 
apply the prior knowledge of clustering prototype, otherwise 
the algorithm will be misleading, thus destroying the 
unsupervised and automated algorithm. 

4. Testing on Matlab 
The tests are carried out under Matlab, We compare the two 

algorithms by the result image.The tests were carried out with 
a number K of clusters systematically worth 2, 3, 5 and 10. 
We observed that the best result,  both from a point of 
character detection view than from a time point of view 
calculation,  generally corresponds to the detection of 2 
clusters. He seems quite logical that this value of K is the most 
adequate,  since it will be seen in all the images tested, that 
there are essentially two objects: the background and the 
characters (numbers and words). 

The first image consists of 130x42 pixels. The characters 
that form the word "study" contain mostly green. The 
background of the image consists of seve ral shades of blue. 
The tests on this image give the following results: 

 



 

72 

 

 

Figure 1. Original image (above), application of Fuzzy C-
Means (left) and K-Means (right) with K=2 

 

The second (75x55 pixels) consists of the numbers “22”. 
We see that the contours of the figures are very blurred and 
the part top of image is slightly dark. 

 

 

 

Figure 2. Original image (above), application of Fuzzy C-
Means (left) and K-Means (right) with K=2 

 

 

Figure 3. Fuzzy C-Means (left) and K-Means (right) with 

K=3 

The dark part of the image makes the use of both different 
algorithms; on the one hand, with K=2 (Figure 2), we do not 
clearly distinguish the upper part of "22" and on the other 
hand, when we increase the number of clusters (example with 
K=3 in Figure 3),  we solve this problem but we then loses 
detection of the lower part of the "22". This example 
illustrates that the objects to be segmented must have a 
sufficiently uniform color so that the detection is possible. 

In general, the two algorithms studied give a good result 
but have two drawbacks: on the one hand, they require the 
prior choice of the number K of clusters, which makes it 
impossible to automate the method ; on the other hand, they 
require computation time often high, due to their iterative 
nature. 

We have seen that these two algorithms turn out to be 
effective when the objects in the image (essentially the 
background and characters) are clearly separated. 

We measured the execution time of the two algorithms for 
each image and it appears that FCM is faster for high 
resolution images and that k means is mo re appropriate in the 
case of low resolution images. Note that the computation time 
is never the same for each application of the algorithms, given 
that it is influenced by the initial position of the centroids, 
which is calculated randomly. 

5. Conclusion 
We tested the Fuzzy C-Means and K-Means segmentations 

on a series of color images, using Matlab.We found that the 
two algorithms require prior knowledge of the number of 
clusters to be determined, which makes it impossible to 
possible automation of the process. By their character 
iterative, they turn out to be inefficient when the number of 
clusters becomes important. 

We can therefore conclude that the Fuzzy C-Means and K-
means algorithms are efficient for the detection of characters, 
but are not appropriate for images containing a large number 
of objects. From a computation time point of view, FCM turns 
out to be more efficient for high resolution images. Resolution 
while K-means is more suitable for images of low resolution. 

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