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NON-LINEAR FILTRATION OF IMAGES BASED ON STOCHASTIC
GEOMETRY1
N.G.Fedotov2, A.V. Moiseev3, L.A.Shulga2, A.S. Kol’chugin2
2Penza
State University, Krasnaya St., 40, Penza, 440017, Russia,
tel.: (8412) 368056, [email protected], [email protected]
3All–Russian Distance institute of finance & economics, ul. Kalinina 33B, Penza, 440052
Russia, email: [email protected]
A new geometrical trace-transformation of images, connected with image scanning on
complex trajectories, is presented. It is shown in the work, that trace-transformation is
not only a source of new class constructive features of image recognition, but is also an
effective mean of pattern preprocessing, preparatory to the recognition procedure. The
result of the trace-transformation could be considered an intermediate image, to which
trace-transformation could be applied once more. Depending on the type of the second
trace-transformation, it is possible to achieve various goals, preprocessing is aimed at,
e.g. it is possible to reduce the noise, segment, smooth, as well as to extract contour or
embossed shell and perform polygonal approximation.
Introduction
In pattern recognition, one traditionally distinguishes a few stages: preparation for recognition (preprocessing), formation of features,
and the decision procedure. In literature on
cybernetics the situation has evolved historically that these stages are considered as entirely separate questions. The approach in terms of
stochastic geometry developed in [1] enabled
research in the stage of feature formation. Results of the features’ theory development are
shown in [2]. The articles mentioned cover the
procedure of generation of new features and
the features’ properties
The peculiarity of the new features (triple features) is that they are structured as a composition of three functionals (F )    (F  l ( ,  )) ,
where p ,  - are normal coordinates of the
scanning line l ( ,  ) , functionals P ,  ,  acting, correspondingly, on variables  ,  ,t . F
stands for the image of the object to be recognized. The result of functional T action is an
image g ( ,  ) on a cylinder or on the Möbius
band. Varying properties of functionals within
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1
This work is supported by RBFR, project No. 05-01-00991
the composition makes it possible to form features with pre-assigned properties. Invariance
and sensitivity of features to motion and linear
deformation of the object, for instance, prove
important.
In this article stochastic geometry methods are
applied to the procedure of non-linear filtration
of images. Thus, the two first stages of pattern
recognition are proposed to be united and performed in one technique, which has hardly ever been mentioned so far.
Suggested methods of achieving the purposes
of the preparatory stage of image recognition
are based on the theory developed in [2],
[3].The general idea that allows to fulfill a
greater number of tasks on the stage of preprocessing, is that the image g ( ,  ) should be
transformed in order to obtain a new function
F ( x, y ) of the image.
The transformation presented in the paper has
been called Dual trace-transformation and is a
generalization of transformation  .
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Dual trace- transformation
Non-liner filtration
Let F ( x, y ) be the function of an image on
plane ( x, y ) . And let us define line l ( ,  , t ) on
the plane, to be assigned with parameters 
Depending on the type of direct and dual tracetransformation, on the stage of preprocessing it
is possible to reduce the noise, segment, or
smooth the mage, as well as to extract contour
or the embossed shell and perform polygonal
approximation.
Trace-transformation is an efficient way to
segment the objects within the image and to
define their number
Fig. 1, а shows an image consisting of three
objects. Trace-matrix of an image consisting
of a few objects, has a characteristic sight, represented in Fig. 1, b.
x  cos  y  sin    ,
(1)
and p; parameter t defines a point on the line.
Let us define the function of two arguments
g ( ,  )  T ( F l ( ,  , t )) as a result of functional
T action, values of variables  and p being
fixed.
According to the functional T action, function
F values on the line l ( ,  ) of plane ( x, y ) result in values of function g ( ,  ) in a point on
plane ( ,  ) .
Transforming (1), we obtain:


x
y
x2  y 2 
cos 
sin     ;
 x2  y 2

x2  y 2


A cos(   )   ,
where A  x 2  y 2 ,   arccos
x
x  y2
2
(2)
.
On the basis of (2), it could be proposed that
function F ( x, y ) value at ( x, y ) point engenders a value of function g on a sinusoid, determined with (2) and situated on plane ( ,  ) .
Consider functional T ( g s( x, y, t )) . s ( x, y, t )
is a sinusoid (2), assigned with parameters x
and y; parameter t defines a point on the sinusoid. Let us define the function of two arguments F ( x, y )  T ( g  s( x, y, t )) as a result
of functional T  action, values of variables x
and y being fixed. Let us name transformation
T  a dual trace-transformation for equations
(1) and (2) being dual.
A consecutive performance of direct and dual
trace-transaction leads to a transaction of image function from F ( x, y ) into F ( x, y ) .
Choosing specific properties of functionals 
and T  makes it is possible to form an equal
transaction as well as to obtain a transaction
with pre-assigned properties.
a)
b)
Fig. 1. Image of several objects: а) initial image;
b) trace-matrix
A separate “wave” of the trace-matrix is corresponded to each object within the image. Thus,
the maximum number of segments cut off the
trace-matrix by a line, parallel to the  axis,
and the number of objects within the image are
equal.
For function n( 0 ) , let us take the number of
intersections of image g ( ,  ) and line    0 .
Functional P could then be presented as
(3)
P( g ( ,  ))  n( ) .
Functional  is defined as functional P maximum value on variable  :
( F )  (P( g ( ,  )))  maxP( g ( ,  )) . (4)

The obtained value of  ( F ) is a feature on
the one hand, but on the other, it represents the
number of objects the image contains.
The segmentation of the image is performed
by drawing lines, which divide the segments of
the image. In a particular case the lines could
be straight.
Consider those enclosed internal areas of the
trace-matrix g ( ,  ) , where the elements of
the matrix take on the value 0. In such areas,
each element with coordinates ( ,  ) restores
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a segmenting straight l with normal coordinates ( ,  ) : l  {( x, y ) : x cos   y sin   } .
Having drawn a straight line from each of the
mentioned areas, one gets an image divided
into a number of smaller ones, none of them
containing more than one object.
For function n( ,  ) , let us take the number of
points of image F and line l ( ,  ) intersections. And let functional T be defined as
(5)
T( F  l )  n( ,  ) .
Let functional T be an integral defined for all
types of admissible values of parameter 
R
P(T( F  l )) 
 T( F  l )d  .
(6)
R
The value of functional  , calculated as in
(4), is the diameter of an object in the image.
It is also possible to measure the area of the
examined object can also be measures by use
of the triple composition of functionals, but
functional T is to be replaced with following:
T( F  l )   f ( ,  , t )dt ,
(7)
F l 
and functional  is to be defined as the first
moment.
Considering functionals (5), (3) and minimum
value of function in the role of  makes it
possible to see whether the objects are located
on one straight. If triple feature has the value
1, it means that the objects are situated on one
straight line.
a)
b)
c)
Fig. 2. Polygonal approximation: a) initial image; b)
trace-matrix; в) result of polygonal approximation
When applying a discrete type of trace–
transformation, functional T is calculated as a
summ of intensity values of all points of the
image, which are located on the scanning line l
The result of such transaction, called a tracematrix, is shown on Fig. 2, b.
Functional (a dual functional) T' is to be defined as:
1, if for all t  S  g ( ,  )  
. (8)
T'( g  S )  
0, if exist t  S : g ( ,  )  
Having applied functional T' (δ value being
fixed at   0 ), we obtain image F' which represents the result of polygonal approximation
of the initial image F. This procedure is illustrated in Fig. 2, c.
Trace-transaction could be described as image
F scanning with use of a number of straight
lines l, whether transaction T  is a process of
scanning the intermediate trace-image g ( ,  )
using curvilinear (sinusoid) trajectories s,
their features being determined with all admissible values of coordinates ( ,  ) .
According to the properties of polygonal approximation, modification of initial image F i
the way shown in Fig. 3, а. does not lead to
changes in reconstructed image F  .
a)
b)
c)
Fig. 3. Polygonal approximation and smoothing :
a) initial image; b) trace-matrix; c) result of polygonal
approximation and effect of smoothing of convex shell
contour
Thus, having performed two trace-transactions
based on (7), (8), but with positive value of
threshold   0 , lead to an effect of smoothing
of convex shell contour. (Fig. 3, c).
In order to mark out the embossed envelope,
let us consider properties of its points. According to direct trace-transformation, a tangent to
the envelope engenders a border point of image g ( ,  ) . On the other hand, according to
the dual trace-transformation, border point of
convex shell engenders a sinusoid on plane
( ,  ) . In order to mark out the shell one
should select all sinusoids containing border
points.
а)
b)
c)
Fig. 4. Convex shell detection: a) initial image; b) convex shell; c) convex shell with smoothed contour
78
Thus, having performed two tracetransactions, we obtain a contour of the embossed envelope of the image. (рис. 4, b) using
 value threshold, different from 0, enables
smoothing of an image’s envelope. It is illustrated in Fig.4, c.
a)
b)
c)
Fig. 5. Noise redaction: a) initial image; b) trace-matrix;
c) reconstruction of convex shell from the noisy image
In the examples above we considered idealized
images. In fact, a recognizing system only operates with images, obtained with certain distortion. (e.g. spatial deformation, different
types of noise, etc.). Noise makes the process
of recognition more complicated. That’s why
development of noise elimination methods
(especially in case they are compatible with
procedure of feature) prove important.
Fig. 5, а. shows a noisy image of an object. In
order to mark out its convex shell, it is necessary to find a type of functional T, which
would sort out elements of the trace-matrix,
corresponding with lines, which actually cross
the initial pattern, and would decrease values
of other elements (Fig. 5, b). A simple example of functional T type that could be successfully used in this situation is one considering
maximum length of segment, obtained as a
result of image and scanning line intersection.
In this case it is reasonable to use the type of
T' functional, which would enable mark out
the convex envelope of the image. The result
of reconstruction of convex shell of a noisy
image is shown in Fig. 5, c.
It is possible to develop better ways of noise
elimination by means of making first functional more complicated. In particular, functionals
containing logarithmic proved to be efficient.
Conclusion
The technique presented has an indisputable
advantage over other approaches. The stages
of preprocessing and generation of image fea-
tures are performed in the same way. A variety
of possible realizations of trace-transactions
makes it possible to solve various goals of
preprocessing.
References
1. Nikolay G. Fedotov. Methods of Stochastic Geometry in Pattern Recognition, Moscow, Radio and
Svayz, 1990 (in Russian).
2. Nikolay G. Fedotov. The pattern recognition features theory based on stochastic geometry// Artificial
intelligence, 2000. - №2. - С. 207-211.
3. Nikolay G. Fedotov, Lyudmila A. Shulga, Alexsandr
V. Moiseev, Feature Generation and Stochastic Geometry // Proceedings of the 4th International Workshop on Pattern Recognition in Information Systems,
PRIS 2004, Porto, Portugal, April 2004, p. 169-175.