The separators Wide Margin (SVMs) and Support Vector Machines in English, are supervised classification methods developed during the 90s by V. Vapnik. The SVMs aim to separate in a space dimension n appropriate, a set of vector data belonging to different classes of linear separators (right, plan or hyper plane). Among the multitude of linear separators that could be used, SVMs seek the most optimal separator, with a maximum distance between classes. According to the severability of data, SVMs are distinguished by two models: the linear SVM (where the data is linearly separable) and non-linear (in case the data are not linearly separable). We have therefore introduced the techniques of learning SVMs to the segmentation classification contained a satellite.
2 SEPARATE THEM WIDE MARGIN
The SVMs have been designed to solve the problems of binary classification supervised, and then extended through the strategies and approaches, the multi-class classification
2.1 The binary separators Wide Margin
Where the data is linearly separable and as part of a binary classification, it is a set of m vector data which is associated labels ti (-1, 1) representative of their class. The linear separator of this data set is defined by
h (x) = wt.x + b = 0
Where w = (w1, ..., wn) is the normal vector,
And x = (x1 ,..., xn) a vector splitter b linearly and the threshold.
The function h (x) will represent the border separating the two classes, the classes' ti another example x is defined as:
ti = sign (+ wt.x b)
For Maximizing the distance between classes through maximizing the margin, we must close distance to vehicles known vectors of support.
As the margin is inversely proportional to the standard w [2],The search for optimal
linear
separator
is:
Min 1 w 2
2
(1)
i:t .h(x ) ≥1
i.
i
The system obtained (1) represents the expression of primal optimization problem in SVMs. In order to simplify the constraints we solve the problem by its dual and using the method of Lagrange, we
get
L
(w,
b,
α):
1
m
2
t
Min
w −∑αi.[(w .xi +b).ti −1]
(2)
2
i=1
≥0 i =1...m
αi
As the αi represent the contribution of a xi element in the design of linear separator h (x), where only the αi corresponding to support vectors are not zero. The Lagrangian term obtained will aim to minimize L (w, b, α) over a: w and b, and maximize L (w, b, α) by providing a: α. So the research on extremum of L (w, b, α):
δL(w,b,α)
m
*
=0 w = αi.ti.xi
(3)
δw
i=1
δL(w,b,α)
m
=0∑αi.ti =0
(4)
δb
i=1
By replacing (3) and (4) in L (w, b, α)
gives
the
following
function:
m
1 m
t
F(α) = αi − ∑t i.t j.αi.αj.x i .x j
(5)
i=1
2 i,j=1
The equation (5) allows us to recreate the primal optimization problem expressed in
(6)
by
its
dual:
m
1
m
t
Max∑αi −
.∑ti..ti.αi.αj.xi.x j
i=1
2 i,j=1
m
=0
α .t
∑i
i
(6)
i=1
α ≥0
i =1..m
i
Thus, the search for optimal returns linear separator has a quadratic programming problem where and αi are reckonable b and w can be deducted.
The function associated decision becomes:
m
t
h (x )= αi.t i.x i .x +b
(7)
i =1
Where the data are not linearly separable, it is usually possible to find a linear separator performing an optimal projection of a space E to another area F larger and using functions called projection Ф (x) as:
φ: E →F
(8)
x =(x1,..,xn ) →φ(x) =(φ1(x),..,φn (x),..)
Avec,CARD(E) <CAED(F)
This allows us to translate the optimization problem expressed (6) by:
m
1 m
t
Max α− . t .t .α.α.x .x
∑i
∑i.
i
i
j
i
j
i=1
2 i,j=1
m
α.t =0
∑i
i
(9)
i=1
0≤α≤C i =1..m
i
This transformation can be done implicitly by the kernel K (x, y) = Ф (x) t.Ф (y) including the following:
3 CORE: 4 generic form: 5 parameters: 6:
Standard
deviation7 :laplacian
polynomial
8:
agenda
polynomial
9:Gaussian
based
radial
10:
standard
deviation
Thus, the optimization problem formulated by (8) relation can be rewritten as follows: In practice it is almost impossible to classify all the data perfectly. That is why
V. Vapnik proposes to introduce new variables known springs, to ease the
constraints,
we
get:
i : ti. h(xi)
≥1-ξι
(10)
The optimization problem becomes:
1
m
Min
w
2
+C ∑ξi
(11)
2
i =1
m
α −
1 m
t
Max
t .t .α.α.φ(x ) .φ(x )
∑i
∑i
j
i
j
i
j
i=1
2 i,j=1
m
=0
(13)
α .t
∑i i
i=1
α ≥0 i=1..m
i
3.
SVMS
MULTI-CLASSES:
As C is a constant adjustment between the margin and errors Thus, the dual expression remains the same (for SVMs linear and nonlinear), the only difference is that all Lagrange multipliers αi must be bounded by the constant Super C.
The application of SVMs to a classification with class k, with k> 2 requires a generalization of SVMs bi-level, an increase of binary classifiers. Among the different approaches that can be used for this purpose there is the approach one against one against one and all.
The dual system optimization in the case of
3.1
The
approach
one
against
One
linear
SVMs
becomes:
m
1 m
This
is to
design every possible
binary
Max∑αi −
∑ti.t j.αi.αj.K(xi,x j)
classifiers, and, for k classes will k. (k-1) /
i=1
2 i,j=1
2 classifiers. To
assign
an
element
to a
m
class
e,
e
must
be
tested
with all
the
∑αi.ti =0
(12)
classifiers
designed,
whenever
e
is
i=1
assigned to a class, we increment a counter
i associated
with
it
(i
is initially
set to
0 ≤αi ≤C i=1..m
zero). e will be awarded to the class that
Where: the table .1 is shown
has
a
meter
at
maximum
value.
3.2 A aproach against all approaches
In this approach and a classification with k
CORE :
generic form :
Parameter
classes, each class i is opposed to k-1 and
:
Laplacian
K (x , y )=exp ( − x −y
δδ:Ecart
other
classes
will
develop
k
binary
type
classifiers. To assign an element to a class
Polynomi
K (x , y )=( x . y +1) p
p :polynô
e, e must be tested against all classifiers
al
me Order
designed,
then
assigned
to
the
class
that
Gaussian
2
K(x, y)=exp(−x −y / 2
δδ:Ecart
presents
a
decision
point
e
maximum.
base
type
radial
4.
APPLICATION
OF
THE
SEGMENTATION SVMS CONTENT
Table .1
The dual system optimization in the case of non-linear SVMs becomes:
OF A SATELLITE IMAGE
We have applied SVMs non-linear approach to one against all the segmentation of the content of satellite
images type Landsat5 TM (Thematic Maps) from the region west Bejaia city dated 15 March 1993 at 9am 45mn. This study area was chosen for its varied landscape may be relevant to the application of SVMs to problems of classification multi-classes. We started our application by loading three images corresponding to the three channels TM1 TM2 and TM3. To ease the use of these images, we first made a contrast enhancement and a composition combining colored blue filter channel TM1, the filter TM3 green channel and red channel filter TM4. On the image we conducted sampling to build our base of learning, the latter representing different classes of pixels identified using knowledge themes:
Mer
Ressac
Sable & sol nu
Céréaliculture
Jachère
Urbain
Brûlis
Maquis
Forêt
Maraîchage
Sebkha 1
Sebkha 2
Figure 1: Identification of classes
Once the basic learning built we conducted our segmentation with the following entries: we set the compromise C to 500 and used a Gaussian kernel as well, we achieved the following results (figure.2):
Figure 2: Images of the resulting test 1 and 3 Gaussian kernel (from left to right)
The first test with a standard deviation of a Gaussian 0.1 gave us the best recognition rate (TR) but also a large number of confusion mainly located at the classroom level Sebkha 2, urban, sand and bare ground. The increase in the
standard
deviation gaussian
level
test 2 and test 3, has resulted in the
decrease of recognition and the
disappearance of
classes Sebkha2,
sand and bare ground. The Tests
Settings
percent
is
shown
as:
Tests
Settings
TR
(%)
1
δ
=
0.1
74.51
2
=
0.5
64.98
3
δ
=
1
62.82
Table 2: Results with the Gaussian kernel
5. SIMULATIONS AND RESULTS
To ensure the recovery of the flow of data without learning sequence equalizer based on maximum likelihood is running a two-step procedure to identify the channel by the algorithm that we described earlier, and then we search the data using the results of this identification.
To assess the performance of the equalizer, we begin by presenting the accuracy of the identification algorithm for different SNR
5-1 Identification of Images of Gaussian kernel:
The identification of the Images of the resulting test 1 and 3 Gaussian kernel (figure.2) is used to detect information symbols.
We worked with a comment that contains ten information symbols (N = 10). The estimated coefficients are represented according to the index iteration, for reports RSB equal to 0, 5 and 10 dB.
