GAUSSIAN KERNEL BASED FUZZY C-MEANS CLUSTERING ALGORITHM FOR IMAGE SEGMENTATION

David C. Wyld et al. (Eds) : CSITY, SIGPRO, AIFZ, NWCOM, DTMN, GRAPHHOC - 2016
pp. 47–56, 2016. © CS & IT-CSCP 2016 DOI : 10.5121/csit.2016.60405
GAUSSIAN KERNEL BASED FUZZY C-MEANS
CLUSTERING ALGORITHM FOR IMAGE
SEGMENTATION
Rehna Kalam1
, Dr Ciza Thomas2
and Dr M Abdul Rahiman3
1
Department of Computer Engineering, Kerala University
rehnakalam@gmail.com
2
Department of Electronics Engineering, Kerala University
cizathomas@gmail.com
3
Department of Computer Engineering, Kerala University
rehman_paika@yahoo.com
ABSTRACT
Image processing is an important research area in computer vision. clustering is an unsupervised
study. clustering can also be used for image segmentation. there exist so many methods for image
segmentation. image segmentation plays an important role in image analysis.it is one of the first
and the most important tasks in image analysis and computer vision. this proposed system
presents a variation of fuzzy c-means algorithm that provides image clustering. the kernel fuzzy
c-means clustering algorithm (kfcm) is derived from the fuzzy c-means clustering
algorithm(fcm).the kfcm algorithm that provides image clustering and improves accuracy
significantly compared with classical fuzzy c-means algorithm. the new algorithm is called
gaussian kernel based fuzzy c-means clustering algorithm (gkfcm)the major characteristic of
gkfcm is the use of a fuzzy clustering approach ,aiming to guarantee noise insensitiveness and
image detail preservation.. the objective of the work is to cluster the low intensity in homogeneity
area from the noisy images, using the clustering method, segmenting that portion separately using
content level set approach. the purpose of designing this system is to produce better segmentation
results for images corrupted by noise, so that it can be useful in various fields like medical image
analysis, such as tumor detection, study of anatomical structure, and treatment planning.
KEYWORDS
CLUSTERING, K-MEANS, FCM, KFCM, GKFCM
1. INTRODUCTION
Image segmentation plays crucial role in many applications, such as image analysis and
comprehension, computer vision, image coding, pattern recognition and medical images analysis.
Many algorithms have been proposed for object segmentation and feature extraction [1]. In this
method, a clustering algorithm for medical and other image segmentation will be considered.
Clustering is useful in several exploratory pattern-analysis, grouping, decision-making, and
machine-learning situations, including data mining, document retrieval, image segmentation, and
pattern classification. However, in many such problems, there is little prior information (e.g.,
statistical models) available about the data, and the decision-maker must make as few assumptions
about the data as possible. It is under these restrictions that clustering methodology is particularly
appropriate for the exploration of interrelationships among the data points to make an assessment
(perhaps preliminary) of their structure. Clustering is a process of partitioning or grouping a given

48 Computer Science & Information Technology (CS & IT)
set of unlabelled objects into a number of clusters such that similar objects are allocated to one
cluster. There are two main approaches to clustering [2].One method is crisp clustering (or hard
clustering) ,and the other one is fuzzy clustering. A characteristic of the crisp clustering method is
that the boundary between clusters is fully defined. However, in many cases, the boundaries
between clusters cannot be clearly defined. Some patterns may belong to more than one cluster. In
such cases, the fuzzy clustering method provides a better and more useful method to classify these
patterns. The FCM employs fuzzy partitioning such that a data pixel can belong to all groups with
different membership grades between 0 and 1.FCM is an iterative algorithm. The aim of FCM is to
find cluster centers (centroids) that minimize objective function. The KFCM is derived from the
original FCM based on the kernel method [3].KFCM algorithm is extended which incorporates the
neighbor term into its objective function [4].Fuzzy clustering is a widely applied method for
acquiring fuzzy patterns from data and become the main method of unsupervised pattern
recognition. Drawback for FCM algorithm is sensitive to noise or outlier. Drawbacks of FCM were
solved by introducing KFCM .In Wu and Gao’s paper [5], the Mercer Kernel based method was
investigated. They proposed the KFCM algorithm which is extended from FCM algorithm. It is
shown to be more robust than FCM .N.A.Mohamed , M.N.Ahmed et al.[6] described the
application of fuzzy set theory in medical imaging. In the proposed system, the content set for the
various MRI real time images is used to calculate the low intensity area in the in homogeneity form
will attain best result for the segmentation and outperforms existing techniques resulting in better
accuracy and predicting factor. This method is applicable in different scale of image for different
orientation in segmenting the images. Intensity In homogeneity images based Clustering approach
is used to overcome the curve in the images, to represent the pure segmented images. Here in
previous approach such as Fuzzy c means it fails to target the clustered set point, which fits in the
imperfect noisy scaled images in the analysis domain, process of imperfection occurrence in the
images due to overlap of the pixel with the different intensity, outcomes lower cluster
segmentation in the minimum level for MRI images. GKFCM clustering approach will calculate
the estimated parameter automatically for the image data. The clustering process is applied in MRI
medical image, for separate group according to their pixel intensity, which is done with the process
called Kernel based Fuzzy C means clustering. Where kernel value is selected based on the
activities of the membership function. Group of features will selected based on the proper tuning
rate of the kernel value, helps in detecting the region separately, level based segmentation is
analyzed to detect the intensity region separately, when it comes to in homogeneity Medical
images, it is a difficult task for this approach to identify the low intensity region, it can be done by
applying the suitable filters to process those images. Initial Impact in clustering of medical images
is the drawback in extracting the biological features and it became difficult in identifying the
clustered region in similar part of the medical images. Reduction of inhomogeneity in the noisy
Medical Images is the extreme end task, and analyzing it feature is open problem and challenging
task which yields less attention of approach, which effect the less segmentation accuracy.
2. LITERATURE SURVEY
2.1 K-Means Algorithm
K-means is one of the simplest unsupervised learning algorithms that solve the well known
clustering problem. The procedure follows a simple and easy way to classify a given data set
through a certain number of clusters (assume k clusters) fixed a priori. The main idea is to define k
centroids, one for each cluster. These centroids shoud be placed in a cunning way because of
different location causes different result. So, the better choice is to place them as much as possible
far away from each other. The next step is to take each point belonging to a given data set and
associate it to the nearest centroid. When no point is pending, the first step is completed and an
early groupage is done. At this point we need to re-calculate k new centroids as barycenters of the

Computer Science & Information Technology (CS & IT) 49
clusters resulting from the previous step. After we have these k new centroids, a new binding has to
be done between the same data set points and the nearest new centroid. A loop has been generated.
As a result of this loop we may notice that the k centroids change their location step by step until no
more changes are done. In other words centroids do not move any more. Finally, this algorithm
aims at minimizing an objective function, in this case a squared error function. The objective
function
2( )
1 1
k n
j
i j
j i
J x c
= =
= −∑ ∑ ,
ADVANTAGES
1) K-Means algorithm is very fast.
2) It is robust and easier to understand.
3) Relatively efficient in the sense it runs in O(tknd) where t is the number of iterations ,k is
the number of
Clusters ,n is the number of objects and d is the dimension of each object.
DISADVANTAGES
1) K-Means algorithm requires a priori specification of the number of cluster centers.
2) If there are two highly overlapping data then k-means algorithm will not be able to
resolve that there are two clusters and is said to be the use of exclusive assignment.
3) It is not invariant to non-linear transformations in the sense we get different results
with different representation of data. Data represented in form of cartesian
co-ordinates and polar co-ordinates will give different results.
4) It provides the local optima of the squared error function.
5) Randomly choosing of the cluster center cannot lead to the good result.
6) Applicable only when mean is defined.
7) Unable to handle noisy data and outliers.
8) It fails for non-linear data set.
2.2.The Fuzzy C Means Clustering Algorithm(FCM)
The fuzzy c-means (FCM) algorithm is one of the most traditional and classical image
segmentation algorithms. The FCM algorithm can be minimized by the following objective
function. Consider a set of unlabeled patterns X, let X={x1,,x2,. ..,xN}, x ∈ Rf, where N is the
number of patterns and f is the dimension of pattern vectors (features). The FCM algorithm focuses

on minimizing the value of an objective function. The objective function measures the quality of
the partitioning that divides a dataset into c clusters. The algorithm is an iterative clustering method
that produces an optimal c partition by minimizing the weighted within group sum of squared error
objective function. Fuzzy c-means (FCM) is a method of clustering which allows one piece of data
to belong to two or more clusters. This method is frequently used in pattern recognition. It is based
on minimization of the following objective function
2
1 1
N C
m
m ij i j
i j
J u x c
= =
= −∑ ∑ ,
where m is any real number greater than 1, uij is the degree of membership of xi in the
cluster j, xi is the ith
of d-dimensional measured data, cj is the d-dimension center of the cluster, and
||*|| is any norm expressing the similarity between any measured data and the center. Fuzzy
partitioning is carried out through an iterative optimization of the objective function shown above,
with the update of membership uij and the cluster centers cj by
2
1
1
1
ij
mc
i j
k i k
U
x c
x c
−
=
=
 −
 
−  
∑
1
1
ij
ij
N
m
i
i
j N
m
j
U x
C
U
=
=
=
∑
∑
ADVANTAGES
1)FCM gives best result for overlapped data set.
2)It is comparatively better than k-means algorithm.
3) Data point is assigned membership to each cluster center as a result of which data point may
belong to more than one cluster center whereas in the case of k-means algorithm data point
must exclusively belong to one cluster center.
DISADVANTAGES
1) FCM requires a priori specification of the number of clusters.
2) Euclidean distance measures can unequally weight underlying factors.
3) We get the better result with lower value of β but at the expense of more number of
iterations.
2.3.The Kernel Fuzzy C Means Clustering Algorithm(KFCM)
The KFCM algorithm adds kernel information to the traditional fuzzy c-means algorithm and it
overcomes the disadvantage that FCM algorithm can’t handle the small differences between
clusters. . The kernel method maps nonlinearly the input data space into a high dimensional feature

space. The essence of kernel-based methods involves performing an arbitrary non-linear mapping
from the original d-dimensional feature space Rd to a space of higher dimensionality (kernel
space). The kernel space could possibly be of infinite dimensionality. The rationale for going to
higher dimensions is that it may be possible to apply a linear classifier in the kernel space while the
original problem in the feature space could be highly non-linear and not separable linearly . The
kernel method then takes advantage of the fact that dot products in the kernel space can be
expressed by a Mercer kernel K. Thus the distance in the kernel space does not have to be explicitly
computed because it can be replaced by a Mercer kernel function (typically referred to as a kernel
trick). There are two major forms of kernel-based fuzzy clustering. The first one comes with
prototypes constructed in the feature space. These clustering methods will be referred to as
KFCM-F (with F standing for the feature space). In the second category, abbreviated as KFCM-K,
the prototypes are retained in the kernel space and thus the prototypes must be approximated in the
feature space by computing an inverse mapping from kernel space to feature space. The advantage
of the KFCM-F clustering algorithm is that the prototypes reside in the feature space and are
implicitly mapped to the kernel space through the use of the kernel function.
3. GKFCM
Given, the GKFCM partitions X into fuzzy subsets by minimizing the
following objective function
Equation 1
c n
2m
m ik k i
i 1 k 1
J (U,V) U x v
= =
= −∑ ∑
Now consider the proposed Gaussian kernel fuzzy c-means (GKFCM) algorithm. Define a
nonlinear map as
Where . X denotes the data space, and F the transformed feature space with higher even
infinite dimension. GKFCM minimizes the following objective function
Equation 2
c n
2m
m ik k i
i 1 k 1
J (U,V) U (x ) (v )
= =
= φ − φ∑ ∑
Where
Equation 3
2
k i k k i i k i(x ) (v ) K(x ,x ) K(v ,v ) 2K(x ,v )φ − φ = + −
Where is an inner product kernel function. If we adopt the Gaussian
function as a kernel function, i.e.,

, then , according to Eqs. (3), Eqs. (2) Can be
rewritten as
Equation 4
( )
1 1
( , ) 2 1 ( , )
c n
m
m ik k i
i k
J U V U k x v
= =
= −∑ ∑
Minimizing Eqs. (4) under the constraint of , we have
Equation 5
( )
( )
1
1
1
1
1
1/ (1 ( , ))
1/ (1 ( , ))
m
k i
ik c
m
k i
j
k x v
u
k x v
−
−
=
−
=
−∑
Equation 6
1
1
( , )
( , )
n
m
ik k i k
k
i A
m
ik k i
k
U K x v x
V
U K x v
=
=
=
∑
∑
Here we just use the Gaussian kernel function for simplicity. If we use other kernel functions, there
will be corresponding modifications in Eqs. (5) and (6).
In fact, Eqs.(3) can be viewed as kernel-induced new metric in the data space, which is defined as
the following
Equation 7
k id(x, y) (x ) (v ) 2(1 k(x, y))= φ − φ = −
CLUSTERING ALGORITHM APPLICATIONS
• Clustering Algorithm can be used in Identifying the Cancerous Data Set.
• Clustering Algorithm can be used in Search Engines.
• Clustering Algorithm can be used in Academics.
• Clustering Algorithm can be used in Wireless Sensor Network ‘s Based Application.
• Clustering Algorithm can be used in Drug Activity Prediction.

4. RESULTS
The experiments and performance evaluation were performed on medical images including a CT
image of the MR image of brain The GKFCM clustering and the pro-posed kernel based fuzzy
level set method were implemented with Matlab R2013a (MathWorks, Natick, MA, USA) in a
Windows 7 System Ultimate. All the experiments were run on a VAIO Precision 340 computer
with Intel i3 and 4GB RAM.
Figure 1 : Original Image
Figure 2 : Cluster 1

5. CONCLUSION
Clustering is one of the efficient techniques in medical and other image segmentation. The primary
advantage of the research work is that it includes the kernel method, the effect of neighbour pixel
information to improve the clustering accuracy of an image, and to overcome the disadvantages of
the known FCM algorithm which is sensitive to the type of noises. The aim of this paper is to
propose a new kernel-based fuzzy level set algorithm for automatic segmentation of medical
images with intensity in homogeneity. It employs Gaussian kernel-based fuzzy clustering as the
initial level set function. It can approximate the boundaries of ROI with parameter estimation
simultaneously well. It provides noise-immunity and preserves image details. It can be useful in
various fields like medical image analysis, such as tumor detection, study of anatomical structure,
and treatment planning
ACKNOWLEDGEMENTS
We thank everyone who helped us in any way.
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AUTHOR
Rehna Kalam born in 1982 is a full time research scholar at Kerala University. She
received the B.Tech degree in Information Technology from Kerala University in 2005
and the M.Tech degree in Computer Science and Engineering from Anna University,
Coimbatore in 2011.Co Authors are Dr Ciza Thomas, Professor, Department of
Electronics Engineering, College of Engineering, Trivandrum and Dr M Abdul
Rahiman, Professor, Department of Computer Science Engineering, LBS Institute of
Technology for Women, Poojappura, Trivandrum.

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