INTRODUCTION
Since, the beginning of the years 1990, various techniques of cerebral imagery
revolutionized this search while making it possible to see the brain. If these
techniques show us what occurs in the brain during a task without having to
open the brainpan, it is especially thanks to progress of computing. So, with
the progress of the computing, the artificial intelligence (Cornuejols
et al., 2002; Bishop, 2006) experienced a great
development. It forms a whole of techniques of representation and decision (Mitchell,
1997; De Beauville and Kettaf, 2005) making it possible
the machines to simulate a behavior similar to that operated by the human one.
The medical imagery (Ding and Wilkins, 2006) became
an essential tool for the decisionmaking aid (Sharif and
Amira, 2009). From where, need for classification (Alphonso,
2001) as process which assigns an entity with a class starting from a whole
of measurements, by the use of new mathematical methods for the treatment and
interpretation of data (Huang and Laia, 2010). The sector
of the algorithms of training (Cristianini and ShaweTaylor,
2000) remains a very active field of research; for that reason in this study,
the authors propose the Support Vector Machines or Separators with Vast Margin
(SVM) (Abe, 2005; Wang, 2005), exactly,
the multiclass: One Versus Rest (OVR) for their performances in the supervised
learning and the classification which are applied to the brain tumours cellules
(Lukas et al., 2004), diagnosed by Magnetic Resonance
Imagery (MRI) and on sure analysis under the microscope of a sample, to identify
the type and the rank of the tumour. The study developed a machine learning
capable to classify the type of the brain tumour by the use of SVM and presented
the strong points of the method. Precisely, we have to show success of the SVM
applied to brain tumour cells.
SUPPORT VECTOR MACHINES
The support vector machine is a new method for classification of data. This
technique of supervised classification suggested in 1992 by Vladimir VAPNIK
(Vapnik, 1998) results from the statistical theory of
the training. Now, it is an active zone of search for machine learning. The
fundamental idea is to build an optimal learning machine for better generalization,
having:
• 
A definite surface in an optimal way 
• 
In general non linear in the space of entry 
• 
Linear in a dimensional space more raised 
• 
Implicitly defined by a kernel function 
Binary SVM: The principal idea of the binary SVM is to implicitly trace
data with a dimensional space more raised by the intermediary of a kernel function
and to solve a problem of optimization to identify the hyperplane whose margin
is maximum, which separates the examples from training (Vapnik,
1998). The hyperplane is based on the support vector i.e., a whole of examples
of training being on the margin. New examples are classified according to the
side of the hyperplane. The problem of optimization generally is formulating
in a manner and takes account of the inseparable data while penalizing distort
them classifications.
Separable linear case: The solution of the problem of optimization (Burges,
1998) is as follows:
• 
Result of the stage of training: The optimal hyperplane
is: 
Correspond to the closest points, i.e., the support vectors that are no null. They are found by solving the following equation:
•Result of the stage of test: The function of decision is obtained by:
Non separable linear case: The algorithm established for the separable
linear case cannot be used in many real problems. In general, of the noisy data
will make separation linear impossible. For this there, the approach above can
be extended to find a hyperplane that minimizes the error on the training example
(Training error). This approach also indicated under the name of soft margin
hyperplane (Vapnik, 1995). We introduce the slack variable
ξ_{i} the error of the example x_{i} with ξ_{i}>0,
I = 1,..., n. The value of ξ_{i} indicates, where, are x_{i}
before a hyperplane of separation:
• 
If ξ_{i}≥1 then x_{i} is badly classified 
• 
If 0<ξ_{i}<1 then x_{i} is correctly classified,
but is inside margin 
• 
If ξ_{i} = 0 then x_{i} is correctly classified and
is outside the margin or with the band of the margin 
Alternatively, the expression of the function changes from
where, C is the weight of error, it is a parameter to be chosen by the user,
larger C corresponding to assign a higher penalty with the errors, is generally
taken 1/n.
Non linear case: How the methods above can be generalized if the function
of decision is not a linear function of the data?, in the linear case space
used is Euclidean space. Separation by hyperplanes is a relatively poor framework
of classification, bus for much of case; the examples which one wants to separate
are not separable by hyperplanes. To continue to profit from the theory which
justifies the search for hyperplanes of maximum margin while using more complex
surfaces of separation (nonlinear), VAPNIK (Boser et al.,
1992) suggests separating not items x_{i}, but of representatives
this points Xi = φ(x_{i}) in a space F of dimension much larger,
than VAPNIK calls: feature space. One gives oneself a nonlinear function noted
φ, such as φ: R^{m}→F. The separation nonlinear of maximum
margin consists in separating the points X_{i} = φ(x_{i}),
as the function of decision is expressed exclusively using scalar products is
thus, one in practice never uses the function directly, but only the kernel
K is defined by: K(x, x') = φ(x).(x') it is a nonlinear scalar product.
Therefore the problem is solved; it is enough to make a transformation: x_{i}.x_{j}
→φ(x_{j}).φ(x_{j}) in all the formulas of hard
margin by using a type of following kernel (Cornuejols,
2002) function:
The solution of Problem of optimization in the nonlinear case is:
• 
Result of the stage of training: The optimal hyperplane
is: 
α_{i} is found by solving the following equation:
•Result of the stage of test: The function of decision is obtained by:
Multiclass SVM: There are several methods for the multiclass (Platt,
1999) but in this article, we tried out One Versus Rest (OVR) which is implemented
in multiclass SVM. OVR is conceptually the method of simplest SVM multiclass.
Here we build of the binary classifiers of K SVM: classify 1 positive against
all other negative classes, classifies 2 against all other classes..., class
K against all other classes. The combined function of decision of OVR chooses
the class of a sample, which corresponds to the maximum value of the binary
functions of decision of K indicated by the other positive hyperplane (Statnikov
et al., 2005). While thus making, hyperplanes of decision calculated
by K SVM, which call into question the optimality of multiclass classification.
Implementation: The data base is extracted from classification of OMS
(Brucher et al., 1999); the ranks are like degrees
of malignity, indicating the biological behavior. The following diagram summarizes
the essential characters of the tumours gathered according to their rank. In
the atlas used, the various tumour entities are also grouped according to their
rank, which corresponds to the images taken to make our classification by the
two artificial methods. The machine used to carry out these tests is a PC (Pentium
4, 2.80 GHz with 256 MO of RAM). And concerning the programming three essential
tools were used:
• 
C++ Builder 6 under Windows to create a graphic interface
summer developed on the one hand for the pretreatment of the classification
of the medical images, which consists in extracting significant information
in RGB (Red, Green and Blue), which will make the object of the supervised
training and the test, so, the input of the binary SVM and Multiclasses
SVM 
• 
Binary SVM under Linux with SVM Light (Joachims, 1999)
for classification binary classification 
• 
SVM Multiclass (OVR) under Linux is a free tool (free) (Joachims,
1999) which was developed for the scientific research whose problem
of optimization was included 
EXPERIMENTATION
Considering for all this part, following points:
• 
CPU: The time of training measured in seconds 
• 
Tr error: The error on the training set 
• 
Te error: The error on the test set 
• 
The classification rate is the average of the well classified of each
label 
Case 1: binary classification (detection): We began with a binary classification,
which consists in detecting the infected part (tumour) of the not infected part
(healthy) as shown in Table 1. This data was a cerebral IRM,
posterior fossa, whose type of tumour is Ependymoma (Brucher
et al., 1999):
Figure 13 show the variation of the error
rate of training and test according to the kernel chosen and to the parameter
of regularization.
We note that in the Fig. 1, the error on training set and the error on test set will decrease with increasing regularization parameter to stabilize 1E04.
We note that in Fig. 2, the error on training set and the error on test set are null despite the change of the regularization parameter.
We notice in the fig that the error on training set and the error on test set are stable and high compared to Fig. 1 and 2.
Table 1: 
No. of images for each type of tumour in case 1 


Fig. 1: 
Error rate of the polynomial kernel in case 1 

Fig. 2: 
Error rate of the radial kernel in case 1 

Fig. 3: 
Error rate of the tangential kernel in case 1 
After several experiments and according to the comparison made between different Kernels, of which the goal is to minimize the error. We deduce that in the case of detection, the radial function as defined in Table 2 makes it possible to minimize the error on training set and the error on test set with any badly classified element in less time (i.e., 0,1 sec).
Table 2: 
Comparison between Kernels in case 1 

Table 3: 
Rate of classification per class in case 1 

Table 4: 
No. of images for each type of tumour in case 2 


Fig. 4: 
Error rate of the polynomial kernel in case 2 
One passes to the validation results by radial Kernel and regularization parameter higher than 1E05 which gave a satisfactory rate of learning (Table 2). Table 3 represents the rate of classification of the examples well classified and badly classified for each class.
In this case, we note that the classification rate is good even all the examples of the validation set of label 1 have been well learned.
Case 2: classification with 4 classes: In this case, we will use four
various types of classes of frequent tumour cells in the child as can be seen
from Table 4 to explode the number of classes and test the
performance of SVM on machine learning. This table represents the affected number
of images for each class, but also the rank of the tumour (Brucher
et al., 1999).
We also present in this case, three graphs as illustrated that corresponding to the error of training and test according to the parameter of regularization and the selected type of kernel.
We note that in Fig. 4, the error on training set and the
error on test set are minimized when the regularization parameter is less than
or equal to 1E03.

Fig. 5: 
Error rate of the radial kernel in case 2 

Fig. 6: 
Error rate of the tangential kernel in case 2 
Then the value of the error begins to increase until stabilization.
We can say that in Fig. 5, the error on training set and the error on test set are minimized and stable after a regularization parameter greater than or equal to 1E02.
In the case of Fig. 6, with a tangential function, the error on training set and the error on test set are stable and equal to 0,75 despite the change of regularization parameter.
According to the tests carried out for each Kernel (Polynomial, radial and tangential), we retain the Polynomial function as shown in Table 5 which minimizes the error on the basis of test and the base of training in less time.
One validates presented results with the parameters obtained by the tests made before (Kernel Polynomial) and one obtains a table which represents the rate of classification of the examples well classified and badly classified in each class depicted in Table 6.
Case 3: classification with 9 classes: In this case, we will burst our
experimentation in nine classes of cerebral tumour in adults as shown in Table
7.
Table 5: 
Comparison between Kernels in case 2 

Table 6: 
Rate of classification per class in case 2 

Table 7: 
No. of images for each type of tumour in case 3 


Fig. 7: 
Error rate of the polynomial kernel in case 3 
This part was added in this study due to scientific curiosity to confirm the
ability of SVM in machine learning. The following table represents the affected
number of images for each class as well as the rank of the tumour (Brucher
et al., 1999).
There we take again the same type of experimentation on this case, by changing the type of the kernel, as well as the parameter of regularization, we obtain the following Fig. 79.
In Fig. 7, the error on training set and the error on test set are minimized when the regularization parameter is equal to 1E05 and it starts to increase with increasing regularization parameter to stabilize after a regularization parameter equal to 1E+01.
We note that in Fig. 8 that the error on training set and
the error on test set are stable even with the change of the regularization
parameter and lower values obtained in the case of using a polynomial function
(Fig. 7).

Fig. 8: 
Error rate of the radial kernel in case 3 

Fig. 9: 
Error rate of the tangential kernel in case 3 
Table 8: 
Comparison between Kernels in case 3 

Table 9: 
Rate of classification per class in case 3 

The Fig. 9 does not differ from other figures in the use of a tangential function in case 1 and case 2, who does not minimize the error on training set and the error on test set.
Us observe in this case, that it is not any more the Polynomial function which
minimized the error in the case 2, on the other hand it is the same function
which minimized the error in case 1, so, the radial function as defined in Table
8.

Fig. 10: 
Comparison of the rates of classification in the three cases 
Table 10: 
Parameters which minimize the error in the three cases 

We introduce the examples of validation and we obtain a rate of classification as shown in Table 9 higher than that found in the case of four classes and that of two classes.
DISCUSSION
One recapitulated the parameters which minimize the error in three cases of
the SVM as illustrated in Table 10. We notice that:
• 
The kernel which minimizes the error changes in three cases
between radial and polynomial, thus we cannot define it exactly 
• 
The parameter of regularization to be chosen as the classification of
the tumour cells, has to be amounts to 10^{5} whether it is for
the kernel radial or polynomial 
• 
The time of learning is satisfactory 
As a consequence, we do say that the SVM has a capacity to learn, in month
of time with a convergence guaranteed according to our experiment. The following
picture presents the classification rate of the tree cases.
By comparing the rate of classification in three cases by SVM as illustrated in Fig. 10, we discover that the SVM is not really influenced by the number of classes and more we increase the number of class, more we have a better rate of classification.
CONCLUSION
The aim of this research ensues from the need to compare the performances of
the various kernels of SVM in classification. In particular, the necessity of
being able to choose in an automatic way the parameters of the kernel of SVM.
The results noticed in the experimentation, are all obtained by alternation
between several parameters for SVM. This last, is one of the models of classification
to have marked the forms recognition by supplying a statistical theoretical
frame and either connexionnist in the learning theory. View Point of learning
time and guarantee of convergence, we can say that SVM is the method the most
adapted to the classification of the brain tumours. We showed examples of the
success of the SVM and suggest some tracks to go farther: voluminous data base
and comparison with others methods as Neural Networks or hybridization with
NeuroSVM (Mitra et al., 2005).