FUZZY IMAGE SEGMENTATION USING VALIDITY INDEXES CORRELATION

International Journal of Computer Science & Information Technology (IJCSIT) Vol 7, No 3, June 2015
DOI:10.5121/ijcsit.2015.7302 15
FUZZY IMAGE SEGMENTATION USING VALIDITY
INDEXES CORRELATION
Daniel Fernando Tello Gamarra1
1
Department of Control Engineering and Automation, Universidade Federal de Santa
Maria (UFSM), Santa Maria, Brazil
ABSTRACT
This paper introduces an algorithm for image segmentation using a clustering technique; the technique is
based on the fuzzy c means algorithm (FCM) that is executed iteratively with different number of clusters.
Furthermore, simultaneously five validity indexes are calculated and their information is correlated to
determine the optimal number of clusters in order to segment an image, results and simulations are shown
in the paper.
KEYWORDS
Image Segmentation, Validity Indexes, Fuzzy C Means, Indexes Correlation.
1. INTRODUCTION
Image processing is used in applications for different fields like medicine, industry or robotics
([1] [2] [3] [4] [5]). Segmentation is the first step in image processing and is the supports for the
execution of other tasks such as refinement or features identification as referred in [6].
Segmentation is a low-level operation with the objective of dividing homogeneous or
heterogeneous regions. The homogeneous regions with borders belong to objects or part of them
in the images. The approach for segmentation followed in this paper applies Fuzzy logic, there are
some previous efforts executed following this approach, for instance, Lopes in [7] uses an
automatic threshold method based on a fuzziness measure; Cheng in [8] uses principles of fuzzy
c-partition and the maximum entropy to select threshold values for gray-level images.
Other topic that explores this work is the use of validity indexes. Previously, Gamarra in [9]
applied validity indexes to find the optimal number of clusters. Besides, some interesting works
have explored the application of validity indexes for image segmentation. Bensaid in [10]
proposed a new validity index, the partition index (SC), and introduced a new clustering
algorithm, the validity-guided (re) clustering (VGC), in Bensaid´s work the validity index is not
used to find the optimal number of clusters, that is already known; the validity index is used to
improve the quality of partitions generated by a clustering algorithm. The VGC was applied
exclusively for Magnetic Resonance Images (MRI) images. In addition, the work developed for
Velthuizen in [11] applies the VGC algorithm to MRI images for brain tumor segmentation.
This work will explore the application of different validity indexes, which will have their
information correlated in order to find an adequate number of clusters for an image segmentation
task. The remainder of the paper is as follows. the second section explains the different
algorithms used in the paper; the third section presents the proposed algorithm designed for the
image segmentation; in the fourth section the experimental platform setup is detailed; in the fifth
section the experimental results are described; finally, conclusions are drawn in the last section.

16
2. THEORETICAL BACKGROUND
2.1. Fuzzy C Means Algorithm (FCM)
As stated in [12], fuzzy C means is very simple; unsupervised and convergent algorithm. The
algorithm description uses equations and terminology written in [13] and resumed in [9]. Given
the data set X , chose the number of clusters c , between Nc <<1 , N is the number of training
data objects, the weighting exponent 1>m , the termination tolerance 0>ε and the norm-inducing
matrix A , if IA = gives the standard Euclidean norm as stated in [14].
The matrix U is the fuzzy partition matrix and represents the membership of the data samples in
the clusters. The weight ikµ represents the degree of membership of an object in a cluster, and is
contained in the kth column of U . An object belongs in some degree to the different clusters, and
each cluster has a centroid iv , if the number of iterations is l , the first step of the algorithm is to
compute the cluster centroids:
ci
x
v N
k
ml
ik
N
k
k
ml
ik
l
i ≤≤=
∑
∑
=
−
=
−
1,
)(
)(
1
1
1
1
)(
µ
µ
(1)
The second step computes the distances:
),()( )()(2 l
ik
Tl
ikikA vxAvxd −−= (2)
Nkci ≤≤≤≤ 1,1
The third step updates the partition matrix elements:
If
0>ikAd
for
ci ≤≤1
,
,1 Nk ≤≤
∑
=
−
= c
j
m
jkAikA
l
ik
dd
1
)1(2
)(
)(
1
µ
(3)
Otherwise 0)(
=l
ikµ if 0=ikAd and [ ]1,0)(
∈l
ikµ with ∑
=
=
c
i
l
ik
1
1µ .
The algorithm lasts a fixed number of iterations or until is reached the condition
ε<− − )1()( ll
UU .
2.2. Validity Indexes in Fuzzy Clustering
The cluster validity as stated in [15] can be used to search for the optimal number of clusters if
the number of clusters is not known. The formulation followed in this subsection took as a
reference the work developed in [15]. The validity indexes used for this paper are the following
and derived from [16]:
The Partition Coefficient (PC) is defined as follows:
∑∑
= =
=
c
i
n
j
iju
n
PC
1 1
21 (4)

17
The Partition Entropy (PE):
∑∑
= =
=
c
i
n
j
ijij uu
n
PE
1 1
2
log
1 (5)
The partition index (SC) given by the equation:
∑
∑
∑
=
=
=
−
−
=
c
i
ij
c
k
m
iji
ij
N
j
m
ij
vxuN
vxu
SC
1 2
1
2
1
)(
)(
(6)
The Separation index (S) on the contrary of SC uses a minimum-distance separation for validity
partition [10]:
2
,
1
2
1
min
)(
ikki
c
i
ij
N
j
m
ij
vvN
vxu
S
−
−
=
∑∑
= =
(7)
The Xie and Beni´s index defined in [17] as follows:
2
1
2
1
min jjji
c
i
jj
n
j
m
ij
vxN
vxu
XB
−
−
=
≠
= =
∑∑
(8)
3. PROPOSED ALGORITHM
Figure 1 depicts the algorithm flowchart, the algorithm will start using a median filter, as the first
step; the next step will be the application of a clustering technique for image segmentation based
in the Fuzzy C Means algorithm. One of the parameters that are necessary to specify for the FCM
is the number of clusters in which the image data will be divided, different cluster numbers were
tested for the algorithm, beginning with 2 clusters until a predefined number of clusters is
achieved (it was chosen 14 for the experiments in this paper). The different validity indexes that
were defined in the paper will evaluate the partitions created for the Fuzzy C Means algorithm in
every run of the algorithm, based on the analysis and the correlation of information provided for
the validity indexes an optimal number of clusters will be selected.
4. EXPERIMENTAL PLATFORM SETUP
4.1. The Matlab Image Processing Toolbox
For the experiments developed in this paper the matlab image processing toolbox will be
employed, this toolbox has a set of routines that let us handle a lot of computational vision
problems.
4.2. The Fuzzy Clustering Toolbox
For the cluster validity was used the Fuzzy Clustering and Data Analysis Toolbox [18]. The
referred toolbox is a collection of Matlab functions with the purpose of dividing a given data set
into clusters.

18
Figure 1. Proposed Algorithm Flowchart
5. EXPERIMENTAL RESULTS
The different experiments were done with the proposed algorithm in this paper. The parameters
used for the clustering algorithm are: number of clusters = 4, maximum number of iterations =
100 (l ), partition matrix exponent = 2 (m), amount of minimal improvement = 1e-6(ε ).
Figure 2 shows the classic figure of the cameraman, that is clustered using the method proposed
on the paper; figure 3 depicts the results obtained applying the validity indexes to the figure, the
partition coefficient (PC) has a monotonic decreasing starting in cluster number 4, also the
classification entropy (CE) has a monotonic increasing starting in cluster 4. Figure 4 shows that
the partition index (SC) and the separation index (S) decreases slowly from c = 4. Also the Xie
and Beni index reaches its local minimum in a region between c = 3 e c = 4. After an evaluation
the information deliver from SC and S are more useful in comparison with the other indexes with
the same c, it was chosen 4 as the optimal number of clusters.
In Table 1 appears the different values obtained for every cluster validity index adopted and its
variation through the different number of clusters tested. c is the clusters number and PC, CE, SC,
S and XB the initials of validity indexes.

19
(a) (b)
(c) (d)
Figure. 2. Cameraman clustered image with FCM: a) original image; (b) 4 clusters; (c) 8 clusters; (d) 14
clusters
Figure 3. Values of Partition Coefficient and Classification Entropy with the clusters number for the
cameraman image.
Figure 4 Values of Partition Index, Separation Index, and Xie and Beni´s Index with the clusters number for
the cameraman image.

20
Table 1 The Numerical Values of Validity Measures ( Cameraman)
C 2 3 4 5 6 7 8
PC 0.9271 0.8731 0.7926 0.8020 0.7939 0.7872 0.7832
CE 0.1348 0.2351 0.3839 0.3841 0.4114 0.4372 0.4557
SC 0.6546 0.5240 0.5314 0.4808 0.4544 0.4024 0.3809
S 0.0999 0.1195 0.1228 0.1109 0.1085 0.1111 0.1053
10-4
XB 622.39 32.7227 29.3341 117.1989 17.7296 61.406 97.276
C 9 10 11 12 13 14
PC 0.7808 0.7686 0.7586 0.7575 0.7480 0.7470
CE 0.4664 0.4968 0.5214 0.5295 0.5538 0.5626
SC 0.3376 0.3331 0.3111 0.3126 0.2968 0.2925
S 0.0927 0.0911 0.0843 0.0839 0.0800 0.0819
XB 50.509 29.6524 23.3215 15.9614 9.5482 146.39
Figure 5 shows the classic figure of lena, that is clustered using the method proposed on the
paper; figure 6 illustrates the results obtained applying the validity indexes to the figure, the
partition coefficient (PC) has a monotonic decreasing starting in cluster number 3, also the
classification entropy (CE) has a monotonic increasing starting in cluster 3. Figure 7 shows that
the partition index (SC) and the separation index (S) decreases slowly from c = 3. Also the Xie
and Beni index reaches its local minimum at c= 3. After an evaluation of the information deliver
for SC and S for this case is more useful in comparison with the other indexes with the same c, it
was chosen 3 as the optimal number of clusters.
In table 2 are shown the different values obtained for every cluster validity indexes adopted and
its variation through the different number of clusters tested.
(a) (b)
(c)
(d)
Figure 5. Lena clustered with FCM: a) original image; (b) 3 clusters; (c) 8 clusters; (d) 14 clusters

21
Figure. 6. Values of Partition Coefficient and Classification Entropy with the clusters number for lena
image.
Figure 7. Values of Partition Index, Separation Index, and Xie and Beni´s Index with the clusters number
for lena image.
Table 2 The Numerical Values of Validity Measures (Lena)
C 2 3 4 5 6 7 8
PC 0.8373 0.7781 0.7834 0.7761 0.7568 0.7526 0.7434
CE 0.2719 0.3975 0.4169 0.44841 0.5014 0.5189 0.5466
SC 1.7675 1.0729 0.7387 0.6438 0.6065 0.5095 0.4912
S 0.6742 0.7052 0.4121 0.4028 0.3669 0.3254 0.3066
(10-
5)
XB 154.26 700.47 33.967 147.051 142.2324 29.471 29.123
C 9 10 11 12 13 14
PC 0.7348 0.7237 0.7233 0.7118 0.7114 0.7039
CE 0.5709 0.5953 0.6017 0.6302 0.6365 0.6553
SC 0.4380 0.3683 0.3538 03492 0.3526 0.2977
S 0.2767 0.2351 0.2242 0.2197 0.2064 0.1903
XB 15.935 30.769 25.815 24.7709 31.8063 38.095
Figure 8 shows the lama image from the Ground Truth database ([19]),that is clustered using the
method proposed on this paper. Figure 9 shows the results obtained applying the validity index to
the figure, the partition coefficient (PC) has a monotonic decreasing starting with cluster number
4, also the classification entropy (CE) has a monotonic increasing starting in cluster 4. Figure 8
shows that the partition index (SC) and the separation index (S) are fixed constant at c = 4. Also
the Xie and Beni index reaches its local minimum at c = 4. Considering that SC and S are more

22
useful, when comparing different clustering methods with the same c, it was chosen 4 as the
optimal number of clusters. Evidently, this figure contains four clusters and the algorithm proves
this fact.
(a)
(b)
(c) (d)
Figure 8. Llama image clustered with FCM: a) original image; (b) 2 clusters; (c) 4 clusters; (d) 14 clusters.
Figure 9. Values of Partition Coefficient and Classification Entropy with the clusters number for lama
image.
for lama image.

23
Table 3 The Numerical Values of Validity Measures (Llama Image)
C 2 3 4 5 6 7 8
PC 0.9171 0.9455 1,000 0.9543 0.9391 0.8130 0.8054
CE 0.1494 0.1088 0.000 0.0633 0.1003 0.2752 0.3015
SC 0.5005 0.1874 0 0 0 0. 0.
S 0.2630 0.1334 0 0 0 0 0
(10-
5)
XB 1.2897 1.7839 1 1.4793 1.4642 1.3380 1.3304
C 9 10 11 12 13 14
PC 0.8008 0.4900 0.4816 0.4672 0.4275 0.4253
CE 0.3219 0.8333 0.8793 0.8966 0.8958 0.9099
SC 0 0 0 0 0 0
S 0 0 0 0 0 0
XB 1.3258 0.7528 0.7444 0.7297 0.6900 0.6878
Figure 11 shows the cell image from the matlab image processing toolbox, that is clustered using
the method proposed on this paper; figure 12 depicts the results obtained applying the validity
index to the figure, the partition coefficient (PC) has a monotonic decreasing starting in cluster
number 3, also the classification entropy (CE) has a monotonic increasing starting in cluster 3.
From figure 13, it is possible to observe that the partition index (SC) and the separation index (S)
hardly decreases at c = 3. Also the Xie and Beni index reaches its local minima at c= 3. After an
evaluation of the information deliver for SC and S are for this case more useful in comparison
with the other indexes with the same c, it was chosen 3 as the optimal number of clusters.
(a) (b)
(c) (d)
Figure 11. Cell image clustered with FCM: a) original image; (b) 2 clusters; (c) 3 clusters; (d) 12 clusters

24
Figure 12. Values of Partition Coefficient and Classification Entropy with the clusters number for the cell
image.
for the cell image.
Table 4 The Numerical Values of Validity Measures (Cell Image)
C 2 3 4 5 6 7 8
PC 0.9416 0.9449 0.9223 0.9154 0.8951 0.7787 0.7715
CE 0.1090 0.1085 0.1646 0.1841 0.2342 0.4140 0.4272
SC 8.0434 1.1791 1.3021 0.8921 0.8948 0.6984 0.5633
S 0.2649 0.0881 0.0797 0.0584 0.0546 0.0451 0.0374
(10-3)
XB 32.632 8.9606 11.9188 6.99 8.6761 4.7263 13.4435
C 9 10 11 12 13 14
PC 0.7703 0.7670 0.7658 0.7621 0.7424 0.7410
CE 0.4440 0.4566 0.4623 0.4694 0.5468 0.5573
SC 0.5812 0.5901 0.5073 0.4321 0.3905 0.4032
S 0.0369 0.0368 0.0322 0.0280 0.0257 0.0258
XB 13.569 41.9482 7.9208 20.5686 2.667 4.2716
6. CONCLUSIONS
The main discernible contribution of the paper is the correlation of the information provided from
different validity indexes in order to define a determined number of clusters. Another contribution

25
of the paper is the guidance search provided by the algorithm to select the optimal number of
clusters in an unsupervised way. In principle, there is not a validity index that can be totally
accurate, so redundant data of the partitions embedded in the validity measures will let us a better
proposal of the optimal number of clusters of the image. Simulations and experiments are shown.
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26
Author
Daniel Fernando Tello Gamarra was born in Peru in 1976, received his BSc.
Degree in mechanical engineering from Universidad Nacional del Centro del Peru
(UNCP), in Huancayo, Peru, in 1999, and his MSc in electrical Engineering from
Universidade Federal do Espirito Santo (UFES), in Vitoria, Espirito Santo, Brazil, in
2004 and PhD degree in Biomedical Robotics from Scuola Superiore Santa Anna,
Pisa, Italy, 2009. He is Professor at the Universidade Federal de Santa Maria
(UFSM), in the Department of Control and Automation in Santa Maria, Rio Grande
do Sul, Brazil. His current research interest centers on robotics, computational vision
and machine learning.

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