ical 
the 
na 
(5) 
sing 
1ges 
ical 
too, 
lage 
(6) 
S to 
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(7) 
S We 
(8) 
led 
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both 
coat 
Petrosyan, Arakel 
  
In the former case the object of examination is the point of the first image and the problem of searching of parameters of 
transformation permitting to combine the same point of the second image with first one is considered. Thus the parameters 
of transformation can vary from point to point and the basic interest represents a field of vectors circumscribing a 
modifications of this parameter. In case of reconstruction of a flow pattern such parameter is the displacement for a 
particular time term, which is easily resulted in a field of velocities. 
The second regional problem puts by the purpose searching of all-purpose parameters, which allow on a given class of 
transformations to combine area of the images or image bodily. Such problems arise, for example, when it is required to 
spot a position of the camera with respect to exterior subjects. In this case parameters of visible deformation of these objects 
are enumerated in coordinates of the camera and angles assigning its orientation. By default all further reasonings concern 
to a problem of dot overlapping. 
It is clear, that the problem of overlapping is reduced to definition of a vector A = {As, Ar}, assigning transformation 
parameters in (7). It is a standard problem of minimisation, which is solved by a method of the gradient descent. Having 
assumed that A > 0, let decompose a right-hand part of expression (7) in a series on vectorial parameter A. For the sake of 
simplicity we first derive the separate formulas for F* and F**. 
g(F Sx, A9) 2 e(FO(,0))  Vg(x) VFO(x,0) A9 ^... (9) 
F®(g(x), A®) = F®(g(x),0) + VF®(g(x),0) A® + ... (10) 
That in aggregate, taking into account parameters Am to be rather small, gives the general formula to within linear terms: 
fix) = g(x) + VFP?(g(x),0) A® + VF™(g(x),0) Vg(x) VF (x,0) A® (11) 
In the terms of conjugate space: 
E(x) = Nix) + Rx) A) + Six) AO + ex) (12) 
Where &{x) = f(x+a;), (x) 2 g(x*aj) - column vectors circumscribing a point of the image in conjugate space, AC AC - 
column vectors assigning parameters deformation and brightness components of transformation, matrixes: 
Ri(x) = OF ®(g(x+a;),0) / 90A®; (13) 
Si(x) = R(x) Ogu(x+ai)ox; OF (x+a;0)/0A® ; (14) 
and e;(x) designates a vector of errors, which is subject to minimization. 
Having designated for B a block matrix {R, S} and having aggregated A® and A® in a single vector of parameters, we 
shall receive 
rx) - Bi(x)A; 7 ei(x), iz1.N, j=1M, N>M (15) 
According to the Gauss - Markov theorem, the solution of such system with a zero right-hand part gives a vector A, 
minimising e(x) in sense of square-law norm. The visual performance of the formula (15) consists that the matrix B sets 
base of transformation F4 at A 0 in conjugate space, and the parameters A determine decomposition coefficients of a 
radius-vector pairing a points of two images on this basis. Apparently, that each M-dimensional parameter generates M- 
dimensional basis, defining M-dimensional subspace in conjugate space. As the radius-vector r has dimensionality N, the 
expansion represents a projection of this vector to a subspace. Generally, when real transformation is tried to approximate 
by transformation with a smaller amount of parameters, the vector of an error appears to be nonzero. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B7. Amsterdam 2000. 1153