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ISSN: 2076-3328
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IJRIC
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AFFINE INVARIANT DESCRIPTORS AND RECOGNIZING
OF 3D OBJECTS USING STATISTICS METHODS
1
M. ELHACHLOUFI , 1A. EL OIRRAK, 2D. ABOUTAJDINE , 1M.N. KADDIOUI
1
Faculty Semlalia , Department of Informatics, Marrakech, Morocco
2
Faculty of Science, LEESA-GSCM, BP 1014, RABAT
E-mail: elhachloufi@yahoo.fr , oirrek@yahoo.fr , aboutaj@fsr.ac.ma, kaddioui@ucam.ac.ma
ABSTRACT
The increasing number of objects 3D available on the Internet or in specialized databases which require the
establishment of methods to develop description and recognition techniques to access intelligently to the contents of
these objects. In this context, our work whose objective is to present the methods of invariant description [1,2,3] and
recognition of 3D objects based on statistical methods: analysis of data (AD) and the principal component analysis
(PCA). The objective of this studies is to determine an invariant description [4,5] of a 3D object using a coefficients
vector of canonical correlations from the data analysis. This vector is invariant against affine transformation of the 3D
object and recognize the object (s) of a database (3D objects) similar(s) to a given object (query object) using the
descriptor vectors extracted from these 3D objects by the principal component analysis.The 3D objects of this database
are transformations of 3D objects by one element of the overall transformation. The set of transformations considered
in this work is the general affine group. The measure of similarity between two objects is achieved by a similarity
function using the Euclidean distance.
Keywords: Invariants Descriptors, Recognizing, 3D objects, Multiple Regression, Principal Component
Analysis, Affine Transformation.
1. INTRODUCTION
transformations and algebraic invariants [10],
which provide global descriptors, which are
expressed in terms of moments of different
orders.
With the advent of the Internet, exchanges and
the acquisition of information, description and
recognition of 3D objects have been as extensive
and have become very important in several
domains.
On the other hand, the size of 3D objects used on
the Internet and in computer systems has become
enormous, particularly due to the rapid
advancement technology acquisition and storage
which require the establishment of methods to
develop description and recognition techniques
to access intelligently to the contents of these
objects.
In fact, several approaches are used: in terms of
statistical approaches, the statistical shape
descriptors for recognition in general consist
either of calculating various statistical moments
[6] [7] and [8], or of estimating the distribution
of the measurement of a given geometric
primitive, when either deterministic or random.
Among the approaches
by statistical
distribution, we mention the specter of 3D shape
(SF3D) [9] which is invariant to geometric
For
structural
approaches,
approaches
representative of the object segmentation in 3D
plot of land and performances by adjacency
graph are presented in [11] and [12]. Similarly,
Tangelder and al [13] have developed an
approach based on representations by interest
points.
In transform approaches a very rich literature
emphasizes any interest in approaches based
transform Haugh [14], [15] and [16] which
consists in detecting different varieties of
dimension (n-1) immersed in the space.
In the same vein, this work interested to define
new methods for description and recognition of
3D objects using statistics methods.
93
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ISSN: 2076-3328
2.
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12V  Corr (V 1 ,V 2 )  0 ,
METHOD
1
:
AN
AFFINE
DESCRIPTION INVARIANT OF 3D
OBJECTS BY THE CANONICAL
CORRELATION COEFFICIENTS
linear
2
uncorrelated
W
1
k
W1
to
2
2
and
2
W are correlated as possible, i.e.: maximizes
2
2
the quantity available  2  Corr (V , W ) .
And so on...
Consider 3D object represented by a set of
denoted M
and W   2Y
i.e.: 12  Corr (W , W )  0 , witch V
2.1 Representation of the 3D objects
points
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 Pi i 1.......n ,
The canonical analysis product p of pairs of
 ( xi , y j , zi )  , arranged in a
matrix X . Under the action of an affine
transformation, the coordinates ( x, y, z ) are
transformed into other coordinates ( x, y, z ) by
variables where s = 1... p. The variables V are
an orthogonal basis of the space generated
the following procedure:
f : 3  3

 X ( x(t ), y (t ), z (t ))  f ( X )  Y ( x(t ), y (t ), z (t ))
Y  A  X ( x(t ), y (t ), z (t ))  B

particularly the first of them, reflect the linear
connections between two groups of initial
where Pi
With
s
3
A  (aij )i , j 1,2,3
invertible
by X
3
matrix
V
1
vectors.
, W 1  which
j
of variables X
j
(X
j
(decreasing) are called
,W s 
s 1,..., p
(  s ) s 1,....., p
is j column of X) and
of
X
and Y
of
canonical
from these couples.
then the
correlations
k
variables
Figure 4
shows the values of canonical
correlation coefficients are all equal to 1. Figures
3 shows that the canonical variables of origin
objects and its transformed are the same which
leads us to conclude that Y is an affine
transformation of X according to the procedure
of canonical analysis.
Y k ( Y k is kth column), such that V 1
1
and W are correlated as possible, i.e: that
maximizes
the
quantity
available
1  Corr (V 1 ,W 1 ) .
Then we search
where
s
coefficients
th
W  1Y a normalized linear combination of
1
 k  Corr (V k ,W k )
V
V  1 X is a normalized linear combination
1
s
variables. The variables V are W are called
canonical variables. Their successive correlations
We consider two 3D objects X and Y
related by an affine transformation (figure 1 and
2). The calculation of canonical correlation
coefficients from these objects requires
computing the first a pairs of canonical variables
The principle of this analysis is to search at the
first a couple of variables
,
2.3 Results and evaluation
t
two
s
s
over an affine transformation and are all equal to
1 in this case.
X   X 1 , X 2 ,........, X p  and
are
s
canonical correlation coefficients (or canonical
correlations).
The
canonical
correlation
(

)
coefficients
s s 1,....., p are invariant quantities
.
Y  (Y1 , Y2 ,........., Yq )t

k
2.2 The canonical analysis of two vectors:
X and Y
Let
s
. The variables W is e an orthogonal
space generated by Y . The couples V , W
associated with the infinite, and B is a vector
translation in
j
the normed pair
V 2   2 X j as
combination of X
j
being
V
2
a
uncorrelated to V
,W 2 
linear
1
Figure 1: 3D Objet origin
, i.e.:
94
Figure 2: 3D Objet transformed
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  ( , u , v )
is a vector characteristic of
X , so we write :
X    ( , u , v )
Figure 3: Representation of
the canonical variables of 3D
object original as function of
the 3D object transformed
(1)
3.2 Representation of 3D objects
Figure 4: Representation of
the canonical correlation
coefficients
Consider a 3D object represented by a set of
Pi i 1....... n
points
denoted
with
Pi t  ( xi , y j , zi ) , arranged in a matrix X ,
3.
X t   P1 , P2 ,..., Pn  where X t and P t
are the matrix and vector transposed X and P .
i.e:
METHOD 2: RECOGNITION 3D
OBJECT PRINCIPAL COMPONENT
ANALYSIS NORMALIZED
Under the action of an affine transformation, the
coordinates of X are transformed into other
coordinates of Y by the following procedure:
Y   X   where  and  are scalar.
3.1 Principal component analysis
The principal component analysis is a
method factorial analysis of multidimensional
data [18]. It determines a decomposition of a
X
random
vector
with
uncorrelated
components, orthogonal and adjusting to better
distribution of X . In this sense the components
are called principal components and are arranged
in descending order according to their degree of
adjustment.
The calculation of normalized principal
components of the vector is carried out initially
by calculating the covariance matrix as follows:
V
Let f function defined as follows:
a f  f (a ) 
where
vi 
Xui
i
and
are the
have :
Y Y

Then f is called an invariant function.
Remarque:
(a)- If X  Y then X
Y .
(b)We
suppose: Y   X  
then
X f  f ( X )  f (Y )  Y f . According to (a)
2- X  u    u  u  u1 , u2 .....u p 
 and u are eigenvalues and eigenvectors
associated V .
The eigenvalues  and eigenvectors v
and
 a2
 X    ( X   )
Y
var( X   )
(X  X ) (X  X ) X  X



 f (X )
 X
X
 2 var( X )
f (Y ) 
1- Det (V   I )  0    1 , 2 ..... p 

(2)
mean and variance X .
We
Then we passes to extract eigenvalues and
eigenvectors associated to V by the following
process
t
a
where a is un 3D object , a and
1 t
X X , Where X t transposed of X .
n
associated V are
aa
we obtain :
v  (v1 ,..., vn )
 X  Y
f
f
, thus
 (.)
f
is an
invariant against affine transformation.
(c)- According to (2) we concluded that:
n is the number of line
X  X f  X  X
 X  ( , u , v )
of X . The reconstruction of X from vector
  ( , u , v ) is doing as follows
If
X   i  ui  v , we say that the vector
X f   i  ui  vit
f
(3)
is a vector characteristic
of X f , i.e :
t
i
i
i
95
(4)
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then
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 X f  ( X f , X ,  X ) is a descriptor of
descriptor, we calculate the error resulting from
the difference between this vector and that of the
object
(query
object)
to
recognize.
We recognize the object that produced the error
is almost zero.
This operation is illustrated in figure 6.
X in the sense that the invariance check is only
relative to the first component  X and the
f
reconstitution X requires the use of
and
X
E-ISSN: 2076-3336
X , X
f
as shown in equation (4).
Unkhown
object
3.3 Principle of description and recognition
proposed method
Extraction of feature
vector  Y associate
Calculation
of
Y
Y f  f (Y )
for this object
3.3.1 Step 1: learning system
Evaluation of
It can be decomposed into two phases:
extraction of vector descriptor and recognition.
The role of the first phase is to associate each
object to learn the vector descriptor
Y
Database
 X f  ( X f , X ,  X ) . This vector will be used
erri   X i  Y
later in the recognition system.
Registration vector descriptors are Phase 2. The
diagram of learning system is shown in figure 5.
If
erri
0 then this object is
recognized as an affine transformation of
Input object
the object Xi , we write
X f  f (X ) 
X
X X
Evaluate
If not , this object is not recognized as an
affine transformation of the object Xi
X
Extraction of feature vector
X
X i  X fi   X i  X i
Figure 6 : Diagram of recogntion system
f
associate for this object
X  E( X ),  X  var( X )
3.4 Results and evaluation
Consider two 3D objects X (object of a
database) and Y (query object) related by an
affine transformation ( figures 1 and 2).
After extraction of vector descriptors  X of X
 X f  ( X f , X ,  X )
f
Y
f
of Y we move to calculate the error vector
err   X  Y
(5)
Database
According to the graph of the error (figure 9) we
found that err  0 , which verifies the invariant
of this vectors as shown in figure 8 and figure
9.
So Y is an affine transformation of X .
Figure 5 : Learning System
3.3.2 Step 2 : Recognition system
For each characteristic vector (first
component) of the object database vector
96
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Figure 8 :descriptor of 3D
objet origin using PCA
Figure 8: descriptor of
3D objet transformed
using PCA
In the second method, the vector descriptor is
formed by the characteristic vector extracted
using PCA from the object origin and its
transformed, this vector is invariant against an
affine transformation.
The recognition is doing by measuring the
similarity between the descriptor vectors
extracted from the two objects: the request object
and those belonging to a database of objects
using the metric euclidean distance as described
above for each method.
Experimental results show the validity and utility
of these methods.
err   X  Y
4. COMPARISON OF RESULTS
According to figures below we can see that
the error1 - corresponds to the difference
between the invariants of the object origin and its
transformation by an affine transformationcalculated by our methods (figures 9) is smaller
than those obtained by the methods of Fourier
transform and Moments (figures 10 and 11).
In addition, the computing time of our method is
less than the method of Fourier transform and
Moments. This leads us to conclude that our
method applied to these objects is more efficient
than Fourier transform and Moments, in term of
error.
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Figure 10: error1
Figure 11: error1
representation by Fourier
transform
representation by
Moments
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data (AD) and the principal component analysis
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The first method consists of calculating the
canonical correlations coefficients vector from
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conclude that the first object (origin) is an affine
transformation of the second (its transformed).
Figure 9: Representation
of error :
IJRIC
[6]
5. CONCLUSION
In this work we presented two methods of
invariant description and recognition of 3D
objects based on statistical methods: analysis of
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