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Intuitively trying to place correspondences between two images, we are guided by any interior, subjective estimation. Thus 
in different situations the essential role is played by the various factors - colour, topological singularities of figures, 
statistical properties, the texture etc. For various objects we select various variants of matching. The legible mathematical 
criterion permitting to compare a points of the image or images and to select the most approaching couples is necessary for 
the purposes of program handling. 
4 MATHEMATICAL DESCRIPTION 
"Similarity" of two of the points is defined by their neighbourhoods - i.e. any characteristic peculiarities located near to a 
considered point. Distracting from the analysis of topological singularities, mathematically this measure of "similarity" or 
“non- similarity ” can be expressed as: 
Rs = i | fixy)- g(x,y) || dxdy (2) 
Where the integration is carried out on a neighbourhood S of a point (x0, y0), and the norm is picked by the most 
convenient mode for example square-law. If f is identically equal g, i.e. the images in a neighbourhood of a point (x0, y0) 
are identical, RS reverts in 0. As for the real images the module f is bounded above, then at given area of an integration S 
value RS is also restricted. In further we assume, that f belongs to a segment [0, fmax]. 
When we speak that the images will be similar, if one of them to turn (to bias, to add luminosities, to stretch etc.), it 
signifies, that before matching in sense (2) it is necessary one of the images to subject to some transformation. Then 
Rs= [|| fixy) - Fa g(x) || dxdy (3) 
S 
Where FA - some operator circumscribing some transformation of the image, maintaining its property, i.e. translating image 
in the image. In such situation we shall speak, that the images are similar on transformation FA, where A is a set of 
parameters of transformation (on how much luminosity has increased, turn angle etc.). However, in spite of the fact that the 
formulas (2-3) allow quantitatively to estimate a measure "non-similarity" of points of the image, we can not conveniently 
define concept of distance, as image is a point infinite-dimensional of function space. For further operation it is necessary to 
narrow down this class of functions and to proceed to finite-dimensional space, which precisely enough would feature the 
starting image and thus would allow to spot as it is necessary to distort the images for a diminution of a non-similarity 
degree. Most simple but not a unique method of build-up of such space is considered below. 
Let's set a Ó - neighbourhood of a given point x of the image f (x) with the aid of netting function 
&(x) = fix+ aj), la|«8à, i=1.N,  a;and8eR? (4) 
The allocation {g;} in neighbourhoods of a point x can be arbitrary, however for a correct image reconstruction it is 
desirable, that they would distribute in a neighbourhood uniformly enough. Problem of an optimal method of allocation of 
nodal points by itself represents separate interest and will be considered later. In such approximation the concept of 
“proximity” will be stated as 
Rix) = (I (f(x+a;) - g(x+ai) Y )? (5) 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B7. Amsterdam 2000. 1151