special integrating fast generalized fuzzy clustering...
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Special Integrating Fast Generalized Fuzzy Clustering (FGCM) Using the Level Set
Methods for Segmentation of the MRI Images
Niloofar Afshar Ghotli 1 , Majid Mohammadi 2, Mehdi Jafari 3
1. Science and Research University, Computer faculty, Kerman,[email protected]
2. Shahid Bahonar University of Kerman, Kerman, [email protected]
3. Islamic Azad University, Kerman, [email protected]
Abstract- Automatic removal of objects from the background is a very functional topic and also is challenging
research field. Many current methods are faced to problem in complicated structures and it depends more to
initializations that it is usually done manually. Therefore, it does not perform fully automated. In this paper we
intend to provide a method based on level set methods and we use of fast generalized fuzzy clustering (FGCM)
for solving initialization. This method involves spatial features of pixels in clustering and so presents better
results in compared to classical fuzzy clustering. The obtained results indicate of high performance and solving
of initialization problem.
Keywords - Level sets, Clustering, Medical images, Segmentation.
1-Introduction
The main objective of medical image is to divide it to different anatomical structures that remove the desired
components such as blood vessels and liver tumors from the ground. Computerized medical image segmentation
is a challenging problem that it is due to poor segmentation power and low contrast. Moreover, this work will be
harder despite of noise and other artificial elements that happen due to tool limitations, reconstruction
algorithms and the patient movement. Still no general algorithm has been designed for medical image
segmentation. Benefits and drawbacks of the algorithm often change according to the problem under
consideration.
Most segmentation algorithms in practice need to qualified and experienced radiologists to adjust carefully the
segmentation parameters for optimal implementation. Most computer systems are implemented in a semi-
automatic or interactive due to the complexity of the medical image segmentation [1]. Radiologists begin
segmentation and stop it if it is required, and finally, stop the algorithm. Obviously, this method is very
subjective and sensitive. Consequently, ease of manipulation (working under the supervision of a radiologist),
often determines the acceptability of this segmentation algorithm in the clinic [2-4].
Level set methods that operate based on the dynamic implicit relationships and partial differential equations
(PDEs) showed that they are useful for medical image segmentation [4-6].
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However, by applying this method, clinical radiologists and also engineer physicians have often confused of
computational requirements and complex regulations of control parameters. Therefore the most advanced
research currently is in such a way while it increases the quality of segmentation, working with it also be easy
[2, 5, 7-9].
There are also hybrid intelligent systems by using of fuzzy clustering for facilitating level set segmentation
[4,5,8,9]. In summary, these algorithms employ fuzzy clustering for the initial segmentation based on image
intensity and employ level set methods for improving target by tracing the boundary changes. In our previous
work on segmentation of liver tumor [4] showed that fuzzy clustering using specified nearly boundaries of liver
has just not decreased the manual intervention but also has increased the optimization of level set. On the other
hand, Ho and Suri have suggested that deformation of level set by fuzzy clustering be adjusted regional to
reduce problems of caused by sensitivity to noise and poor boundaries [5,8,9]. According to their proposed
work, here, fast generalized fuzzy c-means (FGFCM) is used instead of using the classical fuzzy clustering
method. This method is robust against noise and its obtained results are more accurate than the classical and
similar methods. This clustering method is presented in the next section.
Clustering method FGCM
One of the main problems in classical FCM method is the lack of spatial information. This means that this
method clusters pixels only based on the illuminations and does not pay attention to the location of pixel. This
method makes the FCM method becomes highly sensitive to any kind of additive noise. It means that if there is
a noisy pixel in a homogeneous and heterogeneous level, this single pixel is attributed to a different pixel of its
adjacent pixels and this means unconventional output in the segmentation stage.
Methods that are presented for dealing with this problem, have problems themselves: such as the type of noise
should be predetermined, the number of parameters that must be adjusted manually is large. It has a large range
of parameters. Run Time and time complexity of these methods are high and etc. Method that is described here,
is a method that has not many of the above problems, it means it works in the presence of any noise, the number
of parameters that must be adjusted is low, run time is less in compared to the same methods and finally, the
output of clustering is also more suitable and homogeneous. This method uses of both information related to
illumination and spatial information of each pixel for clustering. This method, which is called the fast
generalized FCM (FGFCM) is expressed as follows:
At first, a factor which is called S is defined for each pixel of image. This factor is combined the spatial relations
and local illumination relations of pixels. It is defined as follows:
Equation (1)
In the above definition, i is the studied pixel and also the pixel that is placed among the local window (local
window is a 3 x 3 frame that passes through all pixels and specifies the neighborhood areas), and pixel j is
available pixels in the neighborhood of pixel i. As you can see the value of this factor is dependent on the other
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two factors, namely Ss and Sg. Ss gives the information related to the positions of neighborhood pixels of a pixel
and is defined as follows:
Equation (2)
Which in the above equation (pi,qi) includes of the location coordinates of i-th pixel. It should be noted the
windows that are defined for concept of neighborhood are as square. Value Ss shows the impact of farness or
closeness of a neighbor on the central pixel. Also
is a scale factor for range Ss and its value is determined
simply despite must be predetermined.
Sg defines the amount of local similarity of grey levels for a frame. This factor is defined as follows:
Equation (3)
Which in the last equation
is as follows:
Equation (4)
Which Xi determines illumination of pixel of frame center and Xj determines illumination of pixels in
neighborhoods Ni. Also NR is the number of existent neighborhoods in a frame. is also a scale factor for
range Sg and plays the same role with . Parameter
is a function that determines the amount of local
illumination density for central pixel and homogeneity or heterogeneity of a frame can be determined with the
help of it. Whatever the amount of this parameter is smaller, the studied frame is more homogeneous and vice
versa.
The value of factor
is selected between the range of 0.5 and 6 and it changes with step length of 0.5 (ie 0.5,
1, 1.5, , 6) which normally would be slightly different ranges.
After computing S for all pixels of an image, an image with dimensions of the original image with name
is
formed as follows:
Equation (5)
Which index i indicates i-th pixel in the image. (In fact, l represents a two-dimensional coordinate). For a better
understanding, is supposed that image is converted to a row and we have faced a vector instead of a matrix. (In
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this case,
determines the number of cells this vector). As before,
represents the neighborhood pixels of
pixel I and Ni also represents the neighborhood of i.
After forming image , clustering cost function that should be minimized is defined as follows:
Equation (6)
Which represents the center of i-th cluster,
indicates fuzzy membership function related to illumination l
in i-th cluster. q shows the number f lightning levels. Another important variable that is seen in the recent
equations is . This parameter obtains in this way that at first the image histogram
are made and the number
of iteration of each illumination is determined. Then
is the number of iteration of illumination l in image . In
this way it is seen that cost function navigates through the entire illuminations instead of navigation on the entire
image, and since the number of illumination levels is often less than the number of pixels, this means more
speed of this method in comparison to similar methods. Note that by definition
, we always have:
Equation (7)
Which N is the total number of image pixels.
Now, if we differentiate of
than
and
up to this be minimum in comparison to this parameters , we get
the following equations that these are used for updating cluster centers and their membership functions:
Equation (8)
Equation (9)
Therefore FGFCM method can be summarized as following:
1- We determine the number of clusters and the primary value of centers of these clusters (if we do not
have any information about the range of clusters, the cluster center value is selected as random in order
to adjusting in the next steps)
2- We obtain the value S for all pixels.
3- We make the image
through values S.
4- We compute the value of membership functions according to equation (17).
5- We calculate the value of centers of clusters according to equation (18).
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6- We continue the steps 4 and 5 until we get the maximum iteration numbers or the change range in
membership functions or centers of clusters be less than of a threshold.
This clustering method can be used in order to more accurate segmentation of images in both situations that the
image is including of noise and the image has not any noise. Otherwise, operators still need to adjust carefully
set level for optimal segmentation of level set.
2-Proposed Method
Both FCM algorithm and set level are general-purpose computational models that can be applied for problems
with any dimensions. However, if we limit them to the medical image segmentation, we can take advantage of it
for better implementation. The novel level set algorithm performs automatically the initialization and adjustment
of level set segmentation parameter using spatial fuzzy clustering. This employs FGFCM algorithm with spatial
constraints to determine the desired approximate contours in medical image. With taking advantage of flexible
initialization, intensified level set function can be directly modified FGFCM results for transformation. we
consider in a such way that the desired components in results of FGFCM, is as follows: Rk: {rk= nk, n=x×Ny
+y}, then the initial starting of level set function is simply as follows: equation (10)
Where
is a constant for adjustment of Dirac function [10]. Then Dirac function is defined as:
Equation (11) 0)cos(1
2
1
0,0
)(x
x
x
x
Bk is binary image resulted of equation (3) Bk=Rk+1 that b0( (0,1)) is an adjustable threshold. By taking
advantage of spatial fuzzy clustering, bk can estimate in some cases the desired component that can be adjusted
easily by b0.
Furthermore, With a trial and error, several rules have been proposed for segmenting of optimal level set
[11,10,3]. For example, multiplying of time step in the penalty coefficient ( × ) for stable deformation must be
less than 0.25 and parameter C must be greater than 2 , so that whatever value C be greater, then it makes level
set transformation slower. In our tests it was also found that the larger is, often leads to smoother contours and
whatever
be greater, then deformation of set level be faster. However, there is a risk of leakage boundary.
General guidelines listed above, although is useful, but it is not enough for determining of optimal adjustment
for a specific medical image.
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Determination of control parameters for a specific medical image is more interesting; according to the initial
level set function 0 of spatial fuzzy clustering in equation (1), estimation of length l and area by
Equation (13) )()()( ydxd
And equation (14) )()()( ydxdH
is easy that Heavisible function is H( 0):
Equation (15) 0,0
0,1)(H
We observe that the deformation of level set will be faster if the desired component is large. In this case the
ration
Equation (16) = /
Will be great. Therefore it is logical that we determine the time step in form
in the proposed fuzzy level set
algorithm. Penalty coefficient will be set to as follows:
Equation (17) =0.2/
Because the multiplication ( × ) for stable deformation, should be less than 25/0; the primary function of level
set will approximate the real boundaries that it is obtained of fuzzy clustering equation (2). Therefore,
(with a
relatively conservative value) in form of
Equation (18) =0.1
is used for control of topological changes.
Balloon force
takes two roles in transforming of level set. First, its sign determines the progress direction of
level set function: positive for being smaller and negative for developing. Second, whatever
is the larger, level
set grows faster. In the level set standard algorithms, the controller parameter
is often adjusted as total
constant. Otherwise, it is clear that increasing the growth rate of the level set would be effective,if is still far
away from the actual boundary. On the other side, level set function has to be slow as soon as reaches that
boundary. In addition, the level set function must change its direction automatically when it wants to cross the
desired boundary. We found that the initial segmentation of FCM as a quantitative index, is particularly useful
for adjusting the level set deformation.
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The new fuzzy level set algorithm takes membership degree of each image pixel k as distance to target specific
components of Rk. Here, advanced balloon force has been proposed for pulling or pushing dynamic interface
towards the desired objects:
Equation (19) G(Rk)=1-2Rk
The obtained balloon force G(Rk) ( [-1,1]), is a matrix with variable pusher or tensile force in each image pixel.
In other words, level set function is pulled to the desired object regardless its original location. Then, the
evolutionary equation becomes:
Equation (20) )()()(),( kRgGgdivg
There are many practical benefits of the proposed growth. Now, the balloon force can obtain directly from the
spatial fuzzy clustering. Moreover, level set deformation is consistent with the distance to the actual object.
Upon reaching the object, level set function automatically reduces deformation and it is completely depended to
leveler term. As soon as placed here, the level set deformation is automatically stabilized. Another benefit of
this method is to have flexibility in the choice of a relatively large deformation iteration T for correct
segmentation. Without that increase, the operator must be careful about level set deformation in order to avoid
improper or additional segmentation.
Experiments and Evaluation
Experiments and evaluation of performance of this method conducted on a variety of medical images including
carotid artery sonography image [HYPERLINK \1 "CLi06" 13], CT scan of liver tumors [5] and MRI cutting of
brain tissue [HYPERLINK \1 "DLP99" 14]. Both spatial algorithm FCM and the proposed fuzzy level set
method performed by Matlab (MathWorks, Natick, MA) R2007b). All experiments were performed on a Dell
Pentium IV computer, 2.53 GHz and one GB of memory.
First experiment was designed for evaluating the utility of the initial fuzzy clustering for level set segmentation.
This experiment was employed a fast level set algorithm like [19] for curve optimization that the initialization
performed manually with marking, intensity thresholding, spatial fuzzy clustering. Figure (1) shows the
implementation comparison on sonography image.
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Figure 1: Level set segmentation of Carotid artery sonography: (a) manual initialization; (b) The final
segmentation after 1800 iterations with = 0.1, = 5, = -1.5, = 2; (c) The initialization by thresholding (i
:120-250); (d) The final segmentation after 100 iterations with = 0.1, = 5, = 1.5, = 2; (e) The initialization
by spatial FCM; (f) The final segmentation after 100 iterations, with = 0.1, = 5, = -1.5, = 2.
It is clear that due to the weak boundaries and background strong noise, the manual initialization was not leaded
to level set segmentation (Figure 1a and b). In contrast, both intensity thresholding (Fig. 1 c and d) and fuzzy
clustering (Figure 1e and f) have rapidly attracted the dynamic curve towards the desirable boundaries. It should
be noted that heterogeneity of image is leaded to leakage in boundaries in figure 1 (d). In contrast, the proposed
segmentation FCM with spatial constraints, can be corrected this problem significantly (Figure 1 f).
Figure (2) demonstrates the segmentation of tumor image and liver tissue that have obtained from CT scan
through the fast deformation of level set. There are two areas of cancer tissue close to organ boundary.
Segmentation is performed difficulty due to weak and abnormal boundaries. The liver tissue is heterogeneous
because of blood vessels. Again this problem will be challenging for determining optimal optimization and the
level set parameters associated with it. The results of figure (2) indicate that clustering FCM has the best
performance in initialization of level set. However, without proper controller parameters, level set segmentation
is inappropriate (figure 2, f and h) or is additional (figure 2 j and 1).
Figure 3 shows a more difficult case that needs to separate the white material (WM) and gray material (GM) of
MRI image related to a slice of brain tissue. It is clear that GM and WM overlapped and have separated from the
general cut that it partially does the manually initialization inapplicable (Fig. 3 a and g). Intensity thresholding
(Fig. 3 c and i) and fuzzy clustering (Figure 3 e and k) both are useful in this field. However, obtaining a set of
optimal controller parameters is hard for controlling the deformation of the level set and as shown above,
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without proper adjustments, segmentation of level set is even worse than the original fuzzy clustering (Figure 3
d and j, f and I).
In summary, our proposed fuzzy level set algorithm in segmentation of medical images makes possible the
flexible initialization. In this paper, three paradigms of initialization have been evaluated and compared (Figure
1, 2, 3). Manually marking and intensity thresholding are suitable for initialization of level set. In fact, most of
the systems of level set in this paper are consistent with this kind of initialization [HYPERLINK\l "PAY06" 4
]17]. However, boundaries between physiological tissues in medical images are generally weak and
nonsignificant. Due to the heterogeneity of the image and boundaries leakage, it is clear that manually
initialization is not a reliable choice for optimal segmentation of level set, Fig 1 and 2. In addition, the desired
components are often scattered in the entire image and their marking is not easy individually (separately). In
contrast, the intensity thresholding is useful in this case. However, it requires to accurate adjustment for optimal
thresholds that it is difficult for physiological tissues which have been intertwined (figure 3).
Figure 2: Level set segmentation in CT of liver tissues: (a) and (c) manually initialization; (b) and (d) final
segmentation after 500 iterations, with = 0.1, = 5, = -1.5, = 2; (e) and (g) initialization through the
thresholding (e: 50-95; g :95-250); (f) and (h) initial segmentation after 100 iterations, with = 0.1, = 5, = -
1.5, = 2; (i) and (k) initialization through the spatial FCM; (j) and (l) final segmentation after 100 iterations,
with = 0.1, = 5, = 1.5, = 2.
Initial clustering is able to obtain the adaptive of approximate boundaries of the desired potential components
and therefore is good for beginning of image segmentation. However, standard algorithms FCM that is only
related to intensity information, they are not strong enough for medical image segmentation due to noise and
other artifacts. Intensified FGFCM tries to integrate the intensity and spatial information. This algorithm that is
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related to spatial fuzzy clustering has been shown less sensitivity to various types of noise; therefore it is better
to begin transforming of level set for medical image segmentation.
Figure 3: Segmentation of level set in MRI of the brain tissues (WM and GM): (a) and (g) initial segmentation;
(b) and (h) final segmentation after 2000 iterations, with = 0.1, = 5, = -1.5, = 2; (c) And (i) initialization
through thresholding (e: 170-250; g :120-170); (d) and (j) final segmentation after 100 iterations, with = 0.1,
= 5, = 1.5, = 2; (e) and (k) initialization through the spatial FCM; (f) and (l) final segmentation after 100
iterations, with = 0.1, = 5, = 1.5, = 2.
It is better the presented work in this paper be referred to those which are the same with previous information of
metamorphic models [HYPERLINK\l "NPa03" 15]
Unfortunately, it is not a convenient task to obtain previous
information and models which are related to medical image analysis [39]. For example, computerized
segmentation of liver tumors is a complex task, because both the form and intensity of liver tumors change from
model to another model, from one person to another, and even in various pathological processes. Fuzzy
clustering is able to obtain the desired potential components. Therefore it is considered as an effective source of
previous information for segmenting level set.
Conclusions
In this paper, a new fuzzy algorithm of level set was proposed for automatically segmentation of medical
images. This algorithm uses fuzzy clustering as primary function of level set. FGFCM algorithm which
intensified with spatial information can estimate the desired boundaries well. Therefore, deformation of the level
set will start from area that is close to real boundaries. In addition, the new algorithm estimates control
parameters automatically of fuzzy clustering. It reduces manual intervention. Finally, the equation of level set is
modified by variable balloon forces, so that deformation of level set by fuzzy clustering is adjustable regionally.
In other words, deformation of level set is stabilized automatically as soon as it reaches to real boundaries that
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not only eliminates the leakage of boundaries but also reduces manual intervention. All of these improvements
lead to a robust algorithm for medical image segmentation. Evaluating the performance of these algorithms is
performed by a variety of medical images that its results are promising.
Fuzzy level set method that has been proposed in this paper, has obtained of classical Hamilton- Jacobs function
[18] that is deformation of level set subjected to various internal and external forces. Medical image
segmentation can be considered as Mumford-shah problem, so that level set functions have been formulated in
order to minimize energy function for optimal segmentation [19]. The subsequent methods are not sensitive to
initial contours as much as previous methods are. For this study, the use of the proposed procedures in this paper
is interesting for Mumford-shah level set method for medical image segmentation.
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