Petrosyan, Arakel 
  
As while increasing of an amount of nodal points we can more and more precisely approximate values of function (classical 
theorem of the Weierstrass), the formula (5) on its sense becomes to the equivalent integrated formula (2), though the 
numerical values Rs will differ. Thus, points with neighbourhoods similar in indicated sense we shall define as “close on a 
set {a}N ". Generally speaking, the points, close on one set {a}N not necessarily are close on other set {b}N. 
Interpreting (4) as map R? 2 R and vector £; as a vector of N-dimensional Euclidean space accordingly, we gain that (5) 
sets distance between points in this space. 
The representation of the integral images in new space will have the following properties: 
1. asthe image is function of two variables, a representation of the image will be a two-dimensional surface; 
2. byvirtue of boundedness of values of the image, the fashion bodily belongs to a N-cube [0, fmax]; 
3. the representation has the same tangential properties, as function circumscribing the image; 
4. therepresentation of points of the images having identical neighbourhoods, coincide; 
Thus, the sleek initial image corresponds to a sleek restricted 2D-surface in conjugate N-dimensional space possessing 
composite topology. Further it is considered, that the set {a;}, defining build-up of conjugate space, is given, and the ranges 
of representations, i.e. surfaces in conjugate space, are designated as 2; and 2, accordingly. Apparently, that two identical 
images f (x) and g (x) generate identical X, and Z,. Moreover, if f (x) = g (x+s), then 2, and 2, appear to be identical too, 
though the interior coordinates on these surfaces differ. 
Let's assume, that f(x) = g(x) + £(x), where |e(x)| « £44. Then distance Rs, defining resembling of two points of the image 
has the following property: 
|R((x)| < NEmax Rs(x)2 0 at £,, 2 0 (6) 
Le. at continuous transformation of the starting image g(x) its representation also continuously varies. 
5 OVERLAPPING OF THE IMAGES 
In a general sense overlapping of the images implies searching parameters of some transformation, which allows to 
minimise (in an ideal to reduce to 0) value of Rs. Usually class of transformations is determined by a concrete problem and 
is set a priori. Generally, these transformations can be divided into two groups: deformation (image shaping), operating on 
argument f (x), and modification of luminosity (radiometric shift), changing its value. 
fix) = Fa g(x) = F®( g( F(x, As), Ar) (7) 
Where FA is the generalized transformation with vectorial parameter Am, m-1.. M, and F9 and F are deformation and 
brightness components accordingly. In further we shall assume, that the transformation is continuous and sleek. Besides we 
shall assume, that the transformation is identical at Am=0. 
F4-o f(x) = f(x) (8) 
It is necessary to pay attention, that it is possible to bleed two classes of problems on overlapping the images further called 
dot and regional problems of overlapping. As we spoke, set of points assigning a neighbourhood of a point, should admit 
interpolation reconstruction of function in all neighbourhood, the obtained outcomes of transformation can be referred both 
to the point, and to a neighbourhood as a whole. Thus the size of a neighbourhood can be arbitrary, in particular it can coat 
the image bodily. 
  
1152 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B7. Amsterdam 2000.