f: (8) 
p AAp (9) 
p! BAp 
X7. Note that 
ermined. The 
formula (9) is 
of the right 
supposed to 
i| to zero). 
is equivalent 
(10) 
(11) 
|] symmetric, 
atrix B of 
generalized 
(12) 
À1=0 of n-1 
) À2 is given 
(13) 
3) 22=a'x, where x - eigenvector corresponding to Az. 
From the statement follows, that the decision of 
the problem (6) is under the formula 
Ap - Br (14) 
The effective algorithm of the numerical solution of the 
problem — (14), based on triangular Cholecky 
decomposition of B matrix was also offered in [1]. 
3. USING OF AN IMAGE SEGMENTATION IN THE 
CORRESPONDENCE PROBLEM 
Consider the more complicated a priori data: besides the 
original images a result of segmentation of one of the 
images (for example - left) is obtained. The segmentation 
is a result of low-level or semantic analysis of the image. 
Segmentation splits the image into not crossed regions. 
The development of algorithms of segmentation is 
beyond the scope of this work. Therefore it is considered 
that the result of segmentation is given a priori. 
  
  
  
  
  
  
  
  
  
y 
* f g(xy) 
xi x3 
2 » 
X 
x wl 
Left image Right image 
  
  
  
Fig.1 a Designation of coordinates systems in the model 
with preliminary segmentation 
Let us designate s - segmentation of the template, which 
represented by union of n not crossed regions %i 
i=1,.,n: 
zí]n9 ie 
=| Jz (15) 
The model used in case of three regions is schematically 
shown on Fig.1. Taking into account the geometrical 
distortions of images caused by a perspective projection 
and the three-dimensional form of scene objects for each 
region yi we consider its own affine distortion (16). 
X; = A, +A,X + A3) (16) 
y, ^ by t bx t by 
The transformation (16) enters in each region y; a system 
of coordinates (x;y), determined by a vector of 
parameters pi = (@1i, A2, A3 bai, ba, bai). As in a case with 
only one region (see Section 2), the initial approximation 
of a vector of parameters (a*1,0,b*,0,1) and initial 
distribution of intensity g*(x, y) are known. 
4. LINEARIZATION STEP 
Consider linearization of unknown function g(x,y) in 
region xi with respect to g*(x,y) taking into account 
parameters of transformation (16). In the differential form 
expression (16) is 
Ax, 2 Aa, + xAa,; + yAas 
Ay, = Ab, + xAb,, + yAb, 
(17) 
Linearization of g(xi, yi) in region x; yields 
gx, y,) = g' t g, Aa, * g,x Aa, + g, yAd,, * g, Ab, * 
g,xAb, 3 g,yAb, = g, Ap 
(18) 
a =[o V0 8 g sg VB o£ ss O09 
Ap" z [l Aa, Ad,, Ad, Ah, Ab, Abo) . 
1 Aag,, Aa, Aa, Ab, Ab, 5j. 
1 Aa,, Aa,, Aaj, Ab, Ab,, Ab, , ] 
The vector gi of dimension 7n has the following structure: 
first 7(i-1) of components are equal to zero, 7 coefficients 
of linearization further follow, last 7(n-i) of components 
are also equal to zero. The vector Ap of dimension 7n is 
assembled from transformation parameter amendments 
of all areas. 
5. TRANSFORMATION OF A CROSS-CORRELATION 
COEFFICIENT 
In view of the entered splitting on regions (3) can be 
transformed as follows 
197