w(x) = exp -[x-(M+1)/2]° 
and (27) 
_ M1 Mis s. M as oo! 
Also here, the translation between the images is a secondary effect, 
the dominating effect being the window w(x). Again we obtain y = 0, 
from which we conclude that the procedure is useless. 
IMAGE MODEL WITHOUT WRAP-AROUND 
In the previous sections it was found that wrap-around effects must be 
considered when matching images. It was also found that standard 
methods for compensating wrap-around in image matching are useless. 
The approximation leading from (6) to (9) is too bold and the model 
(7), (8), (11) incorrect. We .now construct a model where the 
wrap-around is compensated for: 
fix) = a c(x)g(xtu) + r(xz);. X = 1,2,...,N 
cix) = 1; X= 1,2,...,,M (28) 
c(x) 7 O; X = M+1,M+2,...,N 
Define f and q to be vectors in an N-dimensional space: 
fe (R fis. f00 0.veoy (29) 
1.2 M 7 
glu) = (S 1 U Fa ris 9, (30) 
and define an N-dimensional window matrix C as 
C = 15 (31) 
where the first M elements on the diagonal are equal to one. 
Following the arguments of equations, (2), (3) and (4), the three 
different cases a), b) and c) as matching criteria give the conditions 
M M 
a) L f(x)g(x+u) = a L c(x)g^ (x*u) ( 32) 
x=1 x=1 
M 1 M 2 2 
b) (9/80) E c(x)f(x)g(x*u) 7» /, a (9/90) Lictix)g (x+y) (33) 
x-1 x=1 
M M 2 2 
c) E c(x)f(x)g(xtu) 4 C 2l ECHO ae] (34) 
x=1 x=1 
where the correlation between f(x) and c(x)g(x+u) has been maximized 
in case c). The relations (32), (33) and (34) correspond to those 
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