be inferior te the result in figure 2a. The reason for this is the 
substitution of f(x) for c(x)f(x) in (35). This has no effect on the 
relations (36), as is seen from the scalar product relations 
f.CU"gq - C£.U"q = £.U%g 
r.BD"g - Bf.D'g - f.p"g (37) 
B= CE 
However, although the scalar products in (37) are identical, the terms 
of which they are sums are not. Including B, (36) gives a linear 
system of equations with unknowns XU kz1,2,...,N. This system being 
singular, there will generally exist a definite generalized inverse 
giving a vector configuration similar to figure 2a. However, as this 
particular inverse is not known, there appears to be no way to reach a 
more definite result than that presented in figure 2d. 
Example: Noncyclically displaced Image Window: 
The displacement corresponds to u = 21, while N = 128 and the image 
window is M = 64. The vectors defined in the text following equation 
(21) (with N substituted for M) are given in figure 2d. The maximum 
component of the sum of the row vectors gives y - 21. A mean of all 
values of yu determined from (18) gives y = 21.3 + 1.5, an essentially 
correct result. However, as is seen in figure 2d, there are also other 
possible translations. Although there are no other differences between 
the images than the translation, the method does not appear to give 
acceptable results. 
THE PHASE CORRELATION FUNCTION 
The relation between a cyclic displacement parameter y for two images 
and their Fourier coefficients was discussed in previous sections. In 
the absence of other discrepancies between the images, we have 
r 7 e 2T iuk/M 
k k 
An inverse Fourier transform of the function $ Er a 2wiuk/M will 
/ 
therefore give a delta function with the spike af x = y. In the more 
realistic case with two different images which overlap without 
wrap-around, the amplitude factor will generally differ from one and 
the phase vary in a more complicated way. However, noting that all 
the information concerning the translation is contained in the 
argument of the exponential function, it is natural to define a 
function à, according to 
(38) 
à o GUT / ME (39) 
and obtain the corresponding image d by an inverse Fourier 
transformation. If the effects of wrap-around and noise are small 
enough, the function d will still resemble a delta function to the 
extent that the matching parameter MW can be determined by 
thresholding. The function d was introduced for the two-dimensional 
case in 1975 by Kuglin and Hines (1) under the name of phase 
correlation function. In their paper they claimed it to be remarkably 
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