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150
MODELS OF FLAT AND SPATIAL
IMAGES CONTOURS ON THE BASIS OF THE THEORY OF A COMPLEX
VARIABLE FUNCTION1
R.G. Khafizov2, А.P. Nefyodov2
2Mari
State Technical University, the Regional Reseach Laboratory of Group Point Objects
and Point Scenes Image Prosessing, Lenin Sq., 3, Yoshkar-Ola, Mari El, 424000, Russia,
tel.: 8(8362) 455412. E-mail: [email protected]
Mathematical models of discrete and continuous contours of flat and spatial images on
the basis of the theory of complex variable function are submitted. Representation of
contours of flat and spatial images as functions of complex or quaternion variables accordingly allows to use the theory of complex variable function for their analysis and
processing.
Introduction
Papers [1,2] consider approaches to processing
the images based on processing of contours of
images which contain the information on the
form of the object, its scale and angular position. Contours of images completely characterize their form and allow to create simple analytical descriptions invariant to transferring,
turning and scaling of images. Consideration
of contours of images as complex-valued signals and their representation in linear complexvalued space allows to receive a measure of
affinity of two contours as their scalar product
invariant to transformations of transferring,
turning and scaling. The question of application of the theory of a complex variable as a
base for processing and the analysis of contours of images is of special interest. The theory of analytical functions of a complex variable represents a set of useful mathematical
models [3,4]. Many mathematical theorems
become simpler if valid variables are considered to be a special case of complex variables.
Complex variables are used for the description
of bidimentional vectors as well as bidimentional scalar and vector fields. Besides analytical functions of a complex variable realize
conformal displays of one plane to another.
The purpose of the work given is the analysis
of an opportunity to use the theory of a complex variable for solving the problems of processing of both flat and spatial images contours.
Models of flat images contours
Let the contour of some image is specified on
plane s by the points connected by vectors
n  , n  0,1,..., s  1 . Each point may be assigned a complex coordinate 1  i 2 . So, if
complex number n  1 n  i 2 n is given
for any natural number n  0,1,..., s  1 , one
can consider that the succession is specified:
Γ  n 0 ,s 1   1 n   i 2 n 0 ,s 1 .(1)
If  n  varies, being subjected to some conditions or others, e.g., Γ  n0, s 1 specifies
the image, we deal with a complex variable
specified on plane C of complex variable 
(Fig. 1).
_______________________________________________________________________
1
This work was supported by the Russian Foundation for Basic Research (Project no. 07-01-00058-а)
151
Im
2 2
1
2 3
2 4
2 0
3
3
1
2 1
ic dotted object that results in quaternion signals.
If some point O is chosen in the space and is
assigned to be the starting point of the reference system, having formed a bundle of s
vectors Q  qn0,s1 connecting point O
2
2
4
Γ
0
with the points of set Ξ , it is possible to consider each of these vectors to be a purely vector quaternion
0
2 s 1
 s 1
1 0 1 1
1 2
s 1
1 s  1 1 3
1 4
Re
Fig. 1. Contour Γ on plane C of complex variable 
r
If some point
with coordinate
r   1 r   i 2 r  in contour Γ is directed
to устремить point r  1 with coordinate
i.e.
r  1  1 r  1  i 2 r  1,
r   r  1  0 , 1 r   1 r  1  0 and
 2 r    2 r  1  0 , the number of points
specifying contour Γ tends to infinity, i.e.
s   , and we move to the notion of continuous contour [5]. Continuous contour
Χ  (l )0,L is represented as continuous
qn  q1 n  i  q2 n  j  q3 n  k ,
n  0,1, ..., s  1 ,
where i , j и k are virtual units. Set of points
Ξ  n0, s 1 specified in such a way is
called
a
quaternion
Q  qn0, s 1 .
signal
(QТS)
Spatial contour D  d n0, s 1 is defined as
the first difference of vectors set by virtual
quaternions QTS Q  qn0, s 1 [2]:
closed graph specified on a complex plane:
Χ  (l )0, L  1 (l )  i 2 (l )0, L , (2)
D  d n0, s 1  qn0, s 1 ,
where l is any value in the range from 0 to L ;
L is the contour length; 1 (l )  Re (l ) and
 2 (l )  Im (l ) are real and virtual components of function (l ) . At multiple passing
along the closed graph contour Χ can be represented as periodic function with period L ,
i.е. (l )  (l  L) ,   0,1, 2,... .
Mathematical models in the form of contours
Γ  n0, s 1 and Χ  (l )0,L allow to de-
where qn  qn  1  qn 
 q1 n  1  q1 n  i  q2 n  1  q2 n  j 
scribe object image contours specified on the
plane.
points
connected
by
vectors
d n ,
n  0,1,..., s  1 , each point in the space is given quaternion coordinate i d1  j d 2  k d 3 . If
virtual
quaternion
d n  i d1 n  j d 2 n  k d 3 n is pointed out
for any natural number n  0,1,..., s  1 , it is
possible
to
consider
that
sequence
D  d n0, s 1 is set up and we deal with
Models of spatial images contours
The application of the quaternion apparatus is
considered to be the most perspective for the
description of the form of spatial images. So in
work [2] quaternion signals Q  qn0, s 1
were used for the description of the image of a
spatial object.
It is the ordered in three-dimensional space set
s of points Ξ  n0, s 1 specifying a gener-
(3)
 q3 n  1  q3 n  k .
The starting point of the reference system
which contains differential quaternion qn
is located in the endpoint of the vector specified by quaternion qn , n  0,1,..., s  1 .
If spatial contour D  d n0, s 1 is set by s
quaternion variable specified in space G of
quaternion variable d (Fig. 2).
152
k
d3 1
1
d3 2
d3 0
d3 s 1
d 0
d1
0
d3 3
d 2
2
d s 1
d1 0
3
Conclusion
s 1
d2 3
d2 1
d2 2
d2 0
d2 s 1
j
plex number. The expediency of introduction
of complex representations of quaternions is
explained by the opportunity to use the apparatus of complex numbers for operations with
quaternions and commutative multiplication.
d1 2 d1 1
Fig. 2. Spatial contour D in space
variable d
d1 s  1
d1 3
i
G of quaternion
In case vectors of spatial contour are defined
by full quaternions d  d 0  d1i  d 2 j  d 3k ,
e.g., vectors of elementary векторы
элементарных QTS such as [2]:
  2 

D m  d m n 0, s 1  exp i mn1  j 

  s
 0, s 1
,
m  0,1,..., s  1 ,
we move to the notion ‘hyper contour’. The
function of quaternion variable f d  can be
specified either in the whole G space or only
in some area of G space. Any area has its internal points and the points of its surface or
hyper contour. The internal points are characterized by the fact that not only the point itself
but its vicinity as well belong to the area as a
whole, i.e. point M will be the internal point
of the area, if some small enough hyper sphere
with centre M wholly belongs to this area. The
hyper contour points are not the internal points
of the area, but there are internal points of the
area in the nearest vicinity of a point of a contour. Thus an area is usually considered to be
only the total of internal points of the area. If
hyper contour joins the area, such an area is
known to be ‘closed’.
Arbitrary quaternion d  d 0  d1i  d 2 j  d 3k
can be represented as the sum of two complex
numbers one of which is multiplied by virtual
1 not coinciding with virtual 1 used in a com-
Representation of contours of flat and spatial
images as functions of complex or quaternion
variables accordingly allows to use the theory
of complex variable function for their analysis
and processing.
References
1. Insights into contour analysis and its application to
image and signal processing /Ed. by Y.А. Furman. –
М.: Fismatlit, 2002.
2. Complex-valued and hypercomplex systems in the
problems of multivariate signals processing / Ed. by
Y.А. Furman. – М.: Fismatlit, 2004.
3. Korn G., Коrn Т. Reference book on mathematics. –
М.: Nauka. Major editorial office for phys.-math.
literature, 1977.
4. Smirnov V.I. The course on Higher Mathematics. –
М.: State editorial office for technical-theoretical literature. V. 2, P. 2. ed. 5. – 1951.
5. Khafizov R.G. The analysis of continuous complexvalued signals specifying the contours of flat images
// Bulletin of Kazan State Technical University
named after А.N. Tupolev, 2006. No 4. P. 24-27.