MATCHING CRITERIA 
Given two images, partly containing the same information, we are 
interested in deciding on the geometrical relation between the images, 
A usual situation is that the images have the same scale and 
orientation, while one image is translated relative the other to an 
unknown extent. The matching problem is then to determine the 
translation. A common approach is to assume an ad hoc model 
representing one of the images as a linear function of the other. 
However, no use is then made of the fact that the images are 
represented by grey levels, which are non-negative. A full linear 
model containing both a multiplicative and an additive constant will 
then violate the non-negative property. Accordingly, the additive 
constant is assumed to be zero and the multiplicative constant 
positive. 
In order to discuss possible methods for determining the translation, 
the problem will be limited to one dimension. This is in fact no real 
limitation. As all strings in an image are translated the same amount, 
the image can be projected on the axis parallel to the strings and the 
matching performed on the projections. Given two projections f(x) and 
g(x), xz1,2,...,N, the matching problem is to "determine a number y 
such that 
f(x) "a g(Xtu) t'r(x); "x 1,2,...,M (1) 
u «"N-V 
a»0 
assuming certain properties of the residual function r(x). 
The conditions on y corresponding to a well matched image pair are 
usually introduced through requirements on r(x): 
a) y is determined in such a way that r(x) and g(x+uy) are orthogonal. 
We then have from (1): 
2 M 
g (xtu) + IL r(x)g(x+u); 
1 x-1 X 
r(x)g(xtyu) = 0 
1 
£(x)g(x*u) a 
1 X 
"ira = 
Hm 3 
lu ra x 
X 
or 
g^ (x*y) (2) 
f(x)g(x*u) a 
1 x 
Hm 
i ra = 
X 
b) u is determined in such a way that the energy in r(x) obtains a 
minimum: 
M 
(8/80) E r^ (x) - O 
x=1 
or 
M 1 M 2 
(8/3u) E f£(x)g(x*u) = /. a (3/34) E 4 (xtu) (3) 
x-1 x=1 
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