INT J COMPUT COMMUN, ISSN 1841-9836
Vol.7 (2012), No. 3 (September), pp. 530-539

Brain Tumor Segmentation on MRI Brain Images with Fuzzy Clustering
and GVF Snake Model

A. Rajendran, R. Dhanasekaran

Arthanari Rajendran
Professor and HoD,
Department of Electronics and Communication Engineering,
Sriguru Institute of Technology,
Coimbatore,Tamilnadu, India
E-mail:rajendranav@gmail.com

Raghavan Dhanasekaran
Professor, Director-Research,
Syed Ammal Engineering College,
Ramanathapuram, Tamilnadu, India
E-mail:rdhanashekar@yahoo.com

Abstract: Deformable or snake models are extensively used for medical image seg-
mentation, particularly to locate tumor boundaries in brain tumor MRI images. Prob-
lems associated with initialization and poor convergence to boundary concavities,
however, has limited their usefulness. As result of that they tend to be attracted to-
wards wrong image features. In this paper, we propose a method that combine region
based fuzzy clustering called Enhanced Possibilistic Fuzzy C-Means (EPFCM) and
Gradient vector flow (GVF) snake model for segmenting tumor region on MRI im-
ages. Region based fuzzy clustering is used for initial segmentation of tumor then
result of this is used to provide initial contour for GVF snake model, which then
determines the final contour for exact tumor boundary for final segmentation. The
evaluation result with tumor MRI images shows that our method is more accurate
and robust for brain tumor segmentation.
Keywords: Deformable model; FCM; Segmentation; MRI image; GVF

1 Introduction

The accurate and automatic segmentation of brain tumor on MRI image is of great interest for as-
sessing tumor growth and treatment responses, enhancing computer-assisted surgery, planning radiation
therapy, and constructing tumor growth models. This is very difficult task in existing methods. The exist-
ing methods are divided into region-based and contour-based methods. Region-based methods [1-9] seek
out clusters of pixels that share some measure of similarity. These methods reduce operator interaction
by automating some aspects of applying the low level operations, such as threshold selection, histogram
analysis, classification, etc. They can be supervised or non-supervised. In general these methods take
advantage of only local information for each pixel and do not include shape and boundary information.
Contour-based methods [10-14] rely on the evolution of a curve, based on internal forces and external
forces, such as image gradient, to delineate the boundary of brain structure or pathology. These meth-
ods can also be supervised or non-supervised. In general these methods suffer from the problem of
determining the initial contour and leakage in imprecise edges.

In this paper we propose a method that is a combination of region-based fuzzy clustering method
called Enhanced Possibilistic Fuzzy C-Means (EPFCM) and Gradient vector flow (GVF) snake model to
remove the problems using the capabilities of each one. For example a region-based method can solve
the problem of the initialization of a contour-based method (GVF snake model) and a contour-based
method is able to improve the quality of region-based segmentation at the boundary of objects.

Copyright c⃝ 2006-2012 by CCC Publications



Brain Tumor Segmentation on MRI Brain Images with Fuzzy Clustering and GVF Snake Model531

So the proposed method has two main phases for tumor segmentation on MRI brain images namely,
initial segmentation which is done by a region-based method and final segmentation that is performed by
a boundary-based GVF snake model. We discuss these approaches in the following section.

2 Region-Based Enhanced Possibilistic Fuzzy C-Means (EPFCM)

In this proposed Enhanced Possibilistic Fuzzy C-Means (EPFCM) method, distance metric Di j in
PFCM [15] is modified in such a way that it includes membership, typicality and both local, nonlocal
spatial neighbourhood information to overcome the noise effect in MRI brain medical images. This
modified distance metric is incorporated into objective function of PFCM. Then resultant algorithm is
called Enhanced Possibilistic Fuzzy C-means (EPFCM) is obtained for enhanced segmentation results.
Therefore objective function of our proposed EPFCM is defined as follows,

Jm (U,V,T ; X) =
c∑

i=1

n∑
j=1

(
aµmi j + bt

η
i j

)
D2i j +

c∑
i=1

γi

n∑
j=1

(
1 − ti j

)η
(1)

Where, the modified distance metric is given by

D2
(
x j, vi

)
& = &D2i j =

(
1 −λ j

)
d2l

(
x j, vi

)
+λ jd

2
nl

(
x j, vi

)
(2)

c∑
i=1

µi j& = &1 ∀ j,0 ≤ µi j, ti j ≤ 1 and a > 0,b > 0,m > 1,η > 1 (3)

The membership function:

µi j =

 c∑
i=1

(
Di j
Dk j

) 2
m−1


−1

(4)

Typicality:

ti j =
1

1 +
(

b
γi

D2i j

) 1
η−1

(5)

Cluster centre:

vi =

n∑
j=1

(
aµmi j + bt

η
i j

)
x j

n∑
j=1

(
aµmi j + bt

η
i j

) (6)
In the following equation is suggested to compute γi:

γi =

K
n∑

j=1
µmi j D

2
i j

n∑
j=1
µmi j

, K > 1 (7)

2.1 Importance of modified distance metric term
(
Di j

)
The modified distance metric or dissimilarity measure is rewritten from Equation (2) as follows,

D2
(
x j, vi

)
& = &D2i j =

(
1 −λ j

)
d2l

(
x j, vi

)
+λ jd

2
nl

(
x j, vi

)
(8)



532 A. Rajendran, R. Dhanasekaran

Where, dl is the distance metric influenced by local spatial information. This added local spatial neigh-
borhood term is similar to the one which is used in [16] to incorporate the neighborhood effects in the
classic FCM.The local spatial constraint is evaluated by the feature difference between neighboring pix-
els in the image.

dnl is the distance measurement influenced by non- local spatial information. This added non local
term is obtained from the non local means (NL-means) algorithm [17] for image denoising. The non-
local constraint determined by all points whose neighborhood configurations look like the neighborhood
of the pixel of interest. λ j is the weighting factor controlling the tradeoff between local and nonlocal
spatial information. It varies from zero to one.

2.2 Importance of local distance metric (Dl)

Let N j denote a chosen local neighborhood configuration of fixed size with respect to a center pixel
x j. If the value of a pixel xk in N j is close to the center pixel, then x j should be influenced greatly
by it, otherwise, its influence to x j should be small. According to the above description, the distance
measurement influenced by local information dl is given by,

d2l
(
x j, vi

)
=

∑
xk∈N j
ωl

(
xk, x j

)
d2 (xk, vi)

∑
xk∈N j
ωl

(
xk, x j

) (9)
where d2 (xk, vi) = ∥xk − vi∥2 is the Euclidean distance metric measure the similarity between pixel pixel
xk and cluster centroid vi,ωl(xk, x j) is the weight of each pixel xk in N j and is given by

ωl
(
xk, x j

)
= e

|xk − x j|2

σ2 (10)

Where, σ2 is the variance of Ni. It specifies the steepness of the sigmoid curve.

2.3 Importance of non local distance metric (Dnl)

The distance measurement influenced by non-local information dnl is computed as a weighted aver-
age of all the pixels in the image I, xk ∈ I

d2nl
(
x j, vi

)
=

∑
xk∈I
ωnl

(
xk, x j

)
d2 (xk, vi) (11)

Where the family of weight ωnl
(
xk, x j

)
; xk ∈ I depends on the similarity between the pixel xk and x j,

and satisfies the usual conditions 0 ≤ωnl
(
xk, x j

)
≤ 1 and

∑
ωnl

(
xk, x j

)
= 1.

The similarity between two pixels xk and x j depends on the similarity of the intensity gray level vector
v(Nk) and v(N j), where Nk denotes a square neighborhood of fixed size and centered at a pixel xk. This

similarity is measured as a decreasing function of the weighted Euclidean distance
∥∥∥∥v(Nk) − v(N j)∥∥∥∥2

2,a
,

where a > 0 is the standard deviation of the Gaussian kernel. The pixels with a similar gray level
neighborhood to v(N j) have larger weights in the average. These weights are defined as

ωnl
(
xk, x j

)
=

1

Q
(
x j

) S (xk, x j) (12)



Brain Tumor Segmentation on MRI Brain Images with Fuzzy Clustering and GVF Snake Model533

Where, S
(
xk, x j

)
is the exponential form of the similarity and Q

(
x j

)
is the normalizing constant. These

terms are defined as,

S
(
xk, x j

)
= e
∥v(Nk )−v(N j)∥22,a

h2 (13)

Q
(
x j

)
=

∑
xk∈I

e
∥v(Nk )−v(N j)∥22,a

h2 (14)

The parameter h acts as a degree of filtering. It controls the decay of the exponential function and
therefore the decay of the weights as a function of the Euclidean distance.

2.4 Importance of trade-off parameter (λ)

For computational purpose, the search of the similar neighborhood configuration always be restricted
in a larger "search window" denoted by Ωi. Let x j be the pixel under consideration. For each pixel xk
in the search window of size S × S , calculate its exponential similarity to x j using Equation (13). The
tradeoff parameter of x j is then defined as

λ j =
1
m

m∑
i=1

S i
(
xk, x j

)
(15)

Where S i represents the ith exponential similarity term in the search window and choose m = S − 1.The
parameter λ j decides the trade-off between local and non local spatial information.

3 Algorithm for Proposed EPFCM Method

Finally the algorithm for carrying out our proposed EPFCM for tumor segmentation of MRI brain
images can now be stated from the following steps

1. Select the number of clusters ′C′ and fuzziness factor ′m′

2. Select initial class centre prototypes v= {vi} ; i = 1,2 · · ·C, randomly and ∈,a very small number

3. Select the neighbourhood size and search window size

4. Calculate modified distance measurement D2i j using the Equation (2)

5. Update membership function µi j using D2i j

6. Update γi; i = 1,2 · · ·C, using Equation (7)

7. Update typicality using the Equation (5)

8. Update cluster centre using equation (6)

9. Repeat steps 4 to 8 until termination. The termination criterion is as follows,

∥Vt+1 − Vt∥ ≤∈ where′t′ is the iteration steps, ∥.∥ is the Euclidean distance norm.

We applied this proposed algorithm to segment tumor on MRI images. In this case, we segmented
the brain image into five classes: namely, CSF (Cerebrospinal fluid), WM (White matter), GM (Gray
matter), tumor and background .Due to some classification errors, there are undesired additional pixels
in the tumor class. To remove these misclassified components, several binary morphological operations



534 A. Rajendran, R. Dhanasekaran

are applied to the tumor class after users defined segmentation classes are obtained (number of clusters).
An opening operation is first used to disconnect the components. Then we select the largest connected
component, which proved to always correspond to the tumor, even if it has a small size. Here the
elementary neighborhood of the morphological operations corresponds to 6-connectivity. The result of
this algorithm gives segmented tumor class as shown in Figure 1(c) .This output is the initial contour for
the GVF snake model.

4 Boundary-Based GVF Snake Model

The traditional deformable active contour model [18-19] is a curve X(S ) =
[
x(s),y(s)

]
, s ∈ [0,1],that

move within the image to minimize the energy function. The curve dynamically changes the shape of
an initial contour in response to internal and external forces. The internal forces provide the smoothness
of the contour. While the external forces push the curve move toward the desired features, such as
boundaries. The object contour will be got when the energy function is minimized. The energy is
defined as:

E =

1∫
0

1
2

[
α|X′(S )|2 +β|X′′(S )|2

]
+ Eext (X(S )) d s (16)

Where, X′(S ) and X′′(S ) are first and second derivatives of X(S ) with respect to s. The parameter α
controls the tension of the curve and β controls its rigidity. Eext is the external energy which is calculated
from the image data. To minimize the energy function, the snake must satisfy the Euler equation

αX′′(S ) −βX′′′′(S ) −∇Eext = 0 (17)

Then the snake is made dynamic by treating as the function of time t, as follows:

Xt (S, t) =αX
′′ (S, t) −βX′′′′(S ) −∇Eext (18)

When the solution X(S, t) stabilizes, the term Xt(S, t) is zero. Then we get the solution of equation
(18).The typical external energies include:

Eext (x,y) = −|∇I (x,y) |2 (19)
Eext (x,y) = −|∇

[
Gσ (x,y) ∗ I (x,y)

]
|2 (20)

Eext (x,y) = I (x,y) (21)

Eext (x,y) = Gσ (x,y) ∗ I (x,y) (22)

Where, Gσ(x,y) is a 2-D Gaussian function with standard deviation σ and mean is zero.∇ denotes
the gradient operator ∗ denotes linear convolution. These external forces have a short capture range and
poor convergence to boundary concavities. To overcome these problems, Gradient vector flow snake was
proposed by Xu and Prince [18], which uses the force balance condition as a starting point of snake. It
defined a new static external force field called GVF field

Fext = V (x,y) = [u(x,y), v(x,y)] (23)

Where, u and v are the grey changes on x-axis and y-axis of the image respectively .Fext can be got by
minimizing the following energy function:

∈=
∫ ∫

µ
(
u2x + u

2
y + v

2
x + v

2
y

)
+ |∇ f |2|v−∇ f |2d xdy (24)



Brain Tumor Segmentation on MRI Brain Images with Fuzzy Clustering and GVF Snake Model535

Where, ux, uy, vx, vy are derivative of x-axis and y-axis respectively. f (x,y) is the edge map (using Canny
edge detector) which is derived from image I(x,y). µ is a regularization parameter governing the tradeoff
between the first term and the second term in the formula. It should be set according to the noise of the
image. The calculus of variations and numerical implementation discussed in [18] is used to obtain the
solution of equation. This deformable contour is first initialized by the tumor class output of EPFCM
method, which then moves towards the final tumor boundary.

5 Results and Discussion

Initially tumor MRI brain image is segmented for tumor class using EPFCM method, which then
initial contour for GVF snake. Then the contour attracted towards final tumor boundary by edge map
derived from the image using Canny edge detector. We set parameter h = 500, search window size is
7 × 7,neighborhood window size is 3 × 3,m = 2,a = 5,b = 3 and η = 2 for EPFCM method to have
proper segmentation result. We set α and β value between 0.1 to 0.2 and µ value between 0.2 to 0.3 for
GVF snake model to have final tumor boundary.The application of our combined method to 10 contrasts
enhanced T1-weighted images and 5 FLAIR images shows better tumor segmentation. The results of
four cases are as shown in Figure 1.

Figure 1: (a)First Column: First two images; Original CE-T1w enhanced tumors; Third image;
Original CE-T1w ring enhanced tumor; Fourth image; Original non enhanced tumor FLAIR image (b)
Second Column: Manual segmentation result (c) Third Column: EPFCM result showing tumor class after
morphological operations (d) Fourth Column: Segmentation of tumor class using combined approach
(EPFCM and GVF snake model) (e) Fifth Column: Final boundary detection (Blue curve) shows tumor
region using GVF snake model.

The evaluation of segmentation performance is also carried out quantitatively by employing four
volume metrics namely, the similarity index(S), false positive volume function (FPVF), false negative
volume function (FNVF) and Jaccard index in our experiment. For a given image, suppose that Ai and
Bi represent the sets of pixels belong to class i in manual and in automatic segmentation, respectively.
|Ai| denotes the number of pixels in Ai. |Bi| denotes the number of pixels in Bi.

The similarity index is an intuitive and clear index to consider the matching pixel between Ai and Bi,



536 A. Rajendran, R. Dhanasekaran

and defined as

S =
2|Ai ∩ Bi|
|Ai|+ |Bi|

(25)

Similarity index S > 70% indicates an excellent similarity [20].
The false positive volume function (FPVF) represents the error due to the misclassification in class

i and the false negative volume function (FNVF) represents the error due to the loss of desired pixels of
class i, they are defined as follows,

FPVF =
|Bi| − |Ai ∩ Bi|

|Ai|
(26)

FNVF =
|Ai| − |Ai ∩ Bi|

|Ai|
(27)

Higher value of S, and lower value of FPVF, FNVF gives better segmentation result.
The Jaccard index between two volumes is represented as follows,

Ji (A, M) =
|Ai ∩ Bi|
|Ai ∪ Bi|

× 100 (28)

Table 1: Evaluation of the segmentation results of enhanced tumors and nonenhanced tumor by com-
bined approach (EPFCM and GVF model) on a few CE-T1w and FLAIR images.(FET denotes the Full
enhanced tumor, RET the ring-enhanced tumor,NET the enhanced tumor

MRI modality type Type of tumor Volume metric functions (%)
CE-T1w & FLAIR FET, RET & NET S FPVF FNVF J

CE-T1w FET1 98.8 0.4 0.2 88.2
CE-T1w FET2 96.3 0.7 0.4 84.5
CE-T1w FET3 92.6 1.2 0.7 86.7
CE-T1w FET4 95.7 0.6 0.6 89.5
CE-T1w FET5 93.2 0.4 0.5 87.8
CE-T1w RET1 95.8 0.1 0.2 80.2
CE-T1w RET2 92.3 1.3 0.7 78.6
CE-T1w RET3 97.6 0.2 0.3 76.2
CE-T1w RET4 91.8 2 1.2 75.5
CE-T1w RET5 96.3 0.1 0.3 77.3
FLAIR NET1 98.8 0.9 0.6 76.9
FLAIR NET2 91.5 2.1 1.8 83.2
FLAIR NET3 98.6 2.9 3 80.1
FLAIR NET4 94.4 1.8 2.2 85.2
FLAIR NET5 95.3 1.2 2.1 81.6

Average 95.3 1.06 0.98 82.1

The result of four volume metrics for our method applied to 15 tumor cases is as shown in Table
1and plotted in Figure 2. From this table, we can see that an average similarity metrics and Jaccard index
of our method is 95.3% and 82.1% that is, the overlap degree between our segmentation result and the
manual segmentation is higher. The average FPVF and FNVF values are equal to 1.06% and 0.98%.
It shows misclassification and loss of desired tumor pixels are reduced in great degree. These average
values are obtained from 15 tumor cases as shown in Figure 3.To compare the results with other methods,
there is no a good standard, however in comparison with works such as in [1,2,7] shows that our method
has a better tumor segmentation performance.



Brain Tumor Segmentation on MRI Brain Images with Fuzzy Clustering and GVF Snake Model537

Figure 2: Graph of the quantitative comparison results of three volume metrics for 15 MRI brain
tumor images.

Figure 3: Graph of the average value of the three volume metrics obtained from 15 MRI brain tumor
images.

6 Conclusions

We have presented in this paper a tumor segmentation method which combines both region based
fuzzy clustering method called EPFCM and boundary based method called GVF snake model .We ver-
ified our method with brain tumour MRI images. The obtained results are quantitatively verified with
other existing methods and they show that our combined approach provides better result.



538 A. Rajendran, R. Dhanasekaran

Bibliography

[1] M. Prastawa, E. Bullitt, S. Ho, G. Gerig, A brain tumor segmentation framework based on outlier
detection, Medical Image Analysis, 2004, 18 (3), 217-231.

[2] J.J. Corso, E. Sharon, A. Yuille, Multilevel segmentation and integrated Bayesian model classifica-
tion with an application to brain tumor segmentation, in: MICCAI2006, Copenhagen, Denmark,
Lecture Notes in Computer Science, October 2006,Vol. 4191, Springer, Berlin, pp. 790-798.

[3] M.B. Cuadra, C. Pollo, A. Bardera, O. Cuisenaire, J. Villemure, J.-P. Thiran, Atlas-based segmen-
tation of pathological MR brain images using a model of lesion growth, IEEE Transactions on
Medical Imaging, 2004,23 (10) ,1301-1313.

[4] J.-P. Thirion, Image matching as a diffusion process: an analogy with Maxwells demons, Medical
Image Analysis, 1998,2 (3) ,243-260.

[5] G. Moonis, J. Liu, J.K. Udupa, D.B. Hackney, Estimation of tumor volume with fuzzy-
connectedness segmentation of MR images, American Journal of Neuroradiology, 2002,23 ,352-
363.

[6] A.S. Capelle, O. Colot, C. Fernandez-Maloigne, Evidential segmentation scheme of multi-echo
MR images for the detection of brain tumors using neighborhood information, Information Fusion,
2004 ,5 ,203-216.

[7] W. Dou, S. Ruan, Y. Chen, D. Bloyet, J.M. Constans, A framework of fuzzy information fusion
for segmentation of brain tumor tissues on MR images, Image and Vision Computing, 2007, 25
,164-171.

[8] M. Schmidt, I. Levner, R. Greiner, A. Murtha, A. Bistritz, Segmenting brain tumors using
alignment-based features, in: IEEE Internat. Conf. on Machine learning and Applications, 2005,
pp. 215-220.

[9] J. Zhou, K.L. Chan, V.F.H Chong, S.M. Krishnan, Extraction of brain tumor fromMR images using
one-class support vector machine, in: IEEE Conf. on Engineering in Medicine and Biology, 2005,
pp. 6411-6414.

[10] A. Lefohn, J. Cates, R. Whitaker, Interactive, GPU-based level sets for 3D brain tumor segmenta-
tion, Technical Report, University of Utah, April 2003.

[11] Y. Zhu, H. Yang, Computerized tumor boundary detection using a Hopfield neural network, IEEE
Transactions on Medical Imaging, 1997 ,16 (1) ,55-67.

[12] S. Ho, E. Bullitt, G. Gerig, Level set evolution with region competition: automatic 3D segmentation
of brain tumors, in: ICPR, Quebec, August 2002, pp. 532-535.

[13] K. Xie, J. Yang, Z.G. Zhang, Y.M. Zhu, Semi-automated brain tumor and edema segmentation
using MRI, European Journal of Radiology, 2005, 56 ,12-19.

[14] Wang Guoqiang, Wang Dongxue, Segmentation of Brain MRI Image with GVF Snake,
Model,in: 2010 First International Conference on Pervasive Computing, Signal Processing and
Applications,2010,pp.711-714.

[15] Pal, N. R., Pal, K., Keller, J. M., and Bezdek, J. C.A, Possibilistic fuzzy c-means clustering algo-
rithm, IEEE Transactions on Fuzzy Systems, 2005, 13(4), pp.517-530.



Brain Tumor Segmentation on MRI Brain Images with Fuzzy Clustering and GVF Snake Model539

[16] Buades A,Coll B,Morel J-M. "A non-local algorithm for image denoising",In CVPR 2005:60-5.

[17] Ma, L. and Staunton, R. C., A modified fuzzy c-means image segmentation algorithm for use with
uneven illumination patterns, Pattern Recognition, 2007, 40(11), pp.3005-3011.

[18] C. Xu and J.L. Prince, Snakes, shapes, and gradient vector flow, IEEE Trans. on Image Processing,
March 1998,vol. 7, pp. 359-369.

[19] Bingrong Wu, Me Xie, Guo Li, Jingjing Gao, Medical Image Segmentation Based on GVF Snake
Model IEEE Conference on Second International Intelligent Computation Technology and Automa-
tion (ICICTA 09), IEEE Press, 2009, vol. 1,Oct., pp. 637 - 640.

[20] Zijdenbos, A. P., Dawant, B. M., Margolin, R. A., and Palmer, A. C. Morphometric analysis
of white matter lesions in MR images: Method and validation, IEEE Transactions on Medical
Imaging, 1994;13(4):716-724.