In this formula NS designates summation on all pixels 
(x,y) 
of the template, N - total number of pixels belonging to 
the template, g - average intensity of the right patch. 
The correspondence problem can be formulated as 
follows: to find 
p* = arg max k(p) (4) 
P 
To take into account the patch shape distortion it is 
offered in [2] to use affine transformation of a kind (5). 
X, =a, + a,x + a,y (5) 
y, = b, + b,x+ by 
For the solving of a problem (4) it is necessary to find a 
vector of parameters (ay,a2,a3,b1,b2, b3)" : 
Suppose that an initial approximation of the parameter 
vector - (a* 1,0,b* 0,1)" is known after first step of 
conventional cross-correlation. Let us denote g*(x,y) 
intensity distribution on the right image patch which 
position is set by an initial vector of parameters. Let us 
denote g*, g, - partial derivatives of g*(x, y). 
Consider linearization of unknown function g(x1,y:) with 
respect to g'(x,y) taking into account parameters of 
transformation (5). 
go, y) eg Ap (6) 
* 
857 E PVP ade i ire 
AP" = [1° Aa, |a, Any Abe Abe AB] - 
vector of the transformation (2) parameter amendments. 
After substitution (6) in (3) 
>. App 
k(Ap) = cx ; e 
dá (2, £^)" (V Ap'gg'Ap - NAp'ggAp)"? 
(x.y) (x.y) 
After equivalent transformations 
196 
k'(Ap) = (k(Ap)) Y. /? (8) 
  
(x,y) 
(4) looks 
Ap” (QD. fg) 9, fg )^p ; 
k'(Ap) = (53) (x) . Ap AAp () 
Ap'(?,gg' - Ngg')Ap  Ap'BAp 
(x.y) 
where 
ARI singular matrix of dimensions 7x7. Note that 
rank of A is equal to one, 
r= > fer - vector of dimension 7, 
(x,y) 
B= > gg’ ~ Nga’ - matrix of dimensions 767. 
(x.y) 
The matrix B - symmetric and positively determined. The 
latter follows due to the denominator of the formula (9) is 
a value proportional to intensity dispersion of the right 
image patch. For real images a matrix B is supposed to 
be non-singular (determinant of B is not equal to Zero). 
Thus (4) is reduced to a problem (10) which is equivalent 
to the generalized eigenvalues problem (11). 
  
s 
Fm 0 
AxzABx (11) 
The following statement was proved in [1]. 
Statement 
Consider any vector a of dimension n and symmetric, 
positively defined and non-singular matrix B of 
dimensions n x n. Then for solutions of a generalized 
eigenvalues problem (11) where 
A=aa" (12) 
the following statements are valid: 
1) There are two generalized eigenvalues: 1,=0 of n-1 
fold and 4250 of 1 fold; 
2) Generalized eigenvector corresponding to 4; is given 
by the formula 
x= B'a (13) 
3) Az-a x, wl 
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