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International Journal of Mathematics and Computer Sciences (IJMCS) ISSN: 2305-7661
www.scholarism.net 4201 April 28Vol.
267
Segmentation of MR Medical Images by Using multi-phase Level Set Method and Bias
Correction
1- SiamakAbdezadeh, Electrical and Computer Engineering Department- Islamic Azad University- Qazvin –
Iran
email: Abdezadeh_b@yahoo.com.
2- FarshidBabapour-Mofrad, Faculty of Engineering, Science and Research Branch, Islamic Azad
University (IAU), Tehran, Iran.
e-mail: Farshid.mofrad@yahoo.com
3- Mohammad Hosntalab, Faculty of Engineering, Science and Research Branch, Islamic Azad University
(IAU), Tehran, Iran.
e-mail: Mhosntalab@yahoo.com
Abstract: In this paper has been proposeda homogeneous method for image segmentation with bias correction in
images.Noise can usually cause inhomogeneity in images andcan lead to incorrect diagnosis. At first, we present a
model for the brightness distribution of each piece in the image and then we transfer it by converting to another
space in which region separation is done better. Finally it is possible toperform simultaneous segmentation and bias
correction by the level set method. Also the proportion method is robust than to initial state selection.
Keywords: Image segmentation, Level set, Bias Correction.
Introduction
Image segmentation is one of the basic images processing that have much applications in recognition affairs and
human machine.Image segmentation plays an important role in the field of analysis and medical images processing
due to the large volume and images. We know the heterogeneity in the brightness intensity of these images with
name bias [3-4].This heterogeneity is due to the lack of appropriate imaging tools andthe effect of magnetic fields on
it. By modeling the images and obtaining the bias, it can be removed that from the image to obtain the better
image.Among the bias correction methods, the methods together with segmentation are highly utilized and
parametric methods that use of the ML and MAP, are also used.
In [2] a method based on simultaneous bias correction and segmentation have been presented that have the
advantages such as robustness compared tothe initial start [9].Since in this method has been used of K-mean
weightedclustering, it is called WKVLS. It can be seen that WKVLS will be a special case of our method. In this
paper we try to implement the optimal values using unspecified starting states by definition of energy function and
using the maximum similarity.
The problem solutionapproach
We use the following method to model the received image:
(1)
https://mail.google.com/mail/h/l20747sx360/?&v=b&cs=wh&to=Abdezadeh_b@yahoo.comhttps://mail.google.com/mail/h/l20747sx360/?&v=b&cs=wh&to=Farshid.mofrad@yahoo.comhttps://mail.google.com/mail/h/l20747sx360/?&v=b&cs=wh&to=Mhosntalab@yahoo.com
International Journal of Mathematics and Computer Sciences (IJMCS) ISSN: 2305-7661
www.scholarism.net 4201 April 28Vol.
267
Where l is the observed image [6], b is bias, n is Gaussian noise with mean 0 and variance sigma and J is also the
original image without noise and bias. If we consider J as the average brightness of that region and equal to ci, then
weindicate each part with Ωiby considering N part of the image. Therefore it can be said that the brightness model
also follows of the Gaussian model with mean bJ and more accurate bci and variance sigma:
(2)
where
(3)
Formulating of the energy function will be in this way. For each point of the image define a neighborhood as
follows:
(4)
Then we transfer the image to another space usinga identity transformation:
(5)
Where
(6)
Because of the transformation, the new space will be also had a Gaussian distribution.
Since the brightness intensity changes smoothly in each part, we can apply the following estimation:
(7)
Also the multiplying of pdfs of Gaussian models would be Gaussian one:
(8)
International Journal of Mathematics and Computer Sciences (IJMCS) ISSN: 2305-7661
www.scholarism.net 4201 April 28Vol.
267
If we define D as follows:
(9)
For similar functionwe have:
(10)
Where:
(11)
If the similarity function is integrated over the entire image, we have:
(12)
The above energy function is obtained.Considering the neighborhood and K definition as follows:
(13)
We have:
(14)
In model WKVLS, the energy function is as follows:
(15)
International Journal of Mathematics and Computer Sciences (IJMCS) ISSN: 2305-7661
www.scholarism.net 4201 April 28Vol.
267
Where Gp is the segmented Gaussian function.It can be seen that if we equal to the specific values sigma and k in
the obtained energy function, then the energy function WKVLS will be achieved. Therefore, model WKVLS is a
special state of energy model that we obtained.
Formulating of level set is as follows:
We use the multi-level set function as in [8]. If we consider that the following function is the characteristic function
of the region ith
(16)
Therefore, the energy function that is called the multilevel statistical function (SVMLS) is defined as follows:
(17)
Where
(18)
In this paper we have taken advantage of minimizing the energy for the four stages. In the fourth stage model we
have:
(19)
Where H is the Heaviside function that we consider it here as usual as follows:
(20)
Minimizing the energy function is performed by considering a variable constantly and minimizing the other. [7-8]
Then we have:
International Journal of Mathematics and Computer Sciences (IJMCS) ISSN: 2305-7661
www.scholarism.net 4201 April 28Vol.
266
(21)
Where
(22)
For minimizing compared to other variable, we have:
(23)
Dirac function has been used as follows:
(24)
It proves that it is enough for staying normalized and uniform the perpendicular vectors to the surfaces, we
convolvethem in a simple Gaussian kernel with limited size.
The main steps of the algorithm are as follows:
1- We select the initial state vectors 1ф and 2ф in such way that they are normalized. It can be considered the
same size and the one in inside and the other in outside of the contour.
2- We update the variables c, b, ϭ by considering the parameters 1ф and 2ф constantly.
3- We update 1ф and 2фby considering variables c, b, ϭconstantly.
4- We stop when the stopping condition has been satisfied.
The output of this step include image segmentation. However, to acquire high accuracy is required to do more
iteration of algorithm that requires much more time to do. for this reason, we use a simpleactive contour model. We
International Journal of Mathematics and Computer Sciences (IJMCS) ISSN: 2305-7661
www.scholarism.net 4201 April 28Vol.
262
form the active contour model with a simple initial state. Then we reverse the result, and multiply it to the result of
the previous step. So we increase the accuracy by using image segmentation hybrid model.
Implementation and results
We have considered in implementation the size of kernel is 3, the level number is 4, and the neighborhood
radius is equal to 4.5. the method result has been compared to with method WKVLS. For both models, the
initial states have been given the same to do the better comparison. Here we have four different levels.
Cerebrospinal fluid, the white section, the gray section and background. Drawing the histogram illustrates that
the image histogram have the stronger peaks and better ability to separate in the proposed model space. The
segmentation results have been also presented. Also the image of MRI has been presented with intensity 7
teslain the following that indicates the available bias has been complicated the diagnosis in image. The result of
bias correction for it has been given in the following.
International Journal of Mathematics and Computer Sciences (IJMCS) ISSN: 2305-7661
www.scholarism.net 4201 April 28Vol.
267
Conclusions
In the proposed method the smoothing of the bias will be able by convolvea simple kernel function. This
simplifies the working with MRI images. Also the proposed model is robust than the initial state. Therefore it
can be utilized for the fully automatic methods. The accuracy has been improved in comparison to the before
methods.
References
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