sum of all the row vectors gives y = 21. L in (18), denoting the 
number of full periods involved in the translation, is now known for 
all different values of k. A mean of all the translations y determined 
from (18) gives u = 21.00 + 0.00 
Example: Noncyclically displaced Sine Curve: 
f(x) = sin 27x/38; x -41,2,...,M 
g(x) = sin 2r(x+21)/38; xm1,2,1:.:,M (23) 
M = 64 
This choice of f and g corresponds to y - 21. The results of matching 
are given in figure 2b. The maximum component of. the sum of the row 
vectors gives y - 33. A mean of all u determined from (18) gives u = 
32.9 + 3.4 . This result is not only very different from the result 
obtained for cyclic translation, it is also seen that any procedure 
for determining u from (18) will be of bad quality. It is concluded 
that the wrap-around error introduced through the definition (10) of 
the unit displacement operator U is disastrous. 
Example: Cyclically displaced Image: 
The displacement corresponds to u - 21, while M = 64. The results of 
matching are given in figure 2a. As is immediately seen, the maximum 
component of the sum of all the row vectors gives u = 21. L in (18), 
denoting the number of full periods involved in the translation, 1s 
now "known for all different values Of k. A mean Of all the 
translations y determined from (18) gives u = 21.00 + 0.00 
Example: Noncyclically displaced Image: 
The displacement corresponds to yu = 21, while M = 64. The results of 
matching are given in figure 2c. The maximum component of the sum of 
the row-vectors gives u = 21. A mean of all values of y determined 
from (18) gives u = 20.8 + 2.4. This result 1s essentially correct 
although it is seen from the bottom part of figure 2c that the 
determination is very uncertain. Any procedure for determining y from 
(18) will be of bad quality. It is concluded that the wrap-around 
error introduced through the definition (10) of the unit displacement 
operator U dominates the result to such an extent that the procedure 
is useless. 
The problem of matching two digital images obtained from photographs 
or any other recording device at two different locations corresponds 
to the case of a noncyclically displaced image. When the images are 
matched using relation (18) without corrections, the results will 
apparently be of bad quality. 
PHASE SHIFT METHOD OF MATCHING 
A tempting approach to the matching problem is to consider sine wave 
components in the image pair and determine the translation y from the 
phase shift between these waves. Given two images f(x), g(x), 
x=1,...,M the sine wave components are obtained as regressions of the 
images on the corresponding sine waves: 
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