power spectra instead of the Fourier transform itself. Following
Ulrych and Jensen (4), the crosspower spectrum between two image
projections can be determined according to
+11 (P
fg bb Peg Pag) aa Pe Pag (40)
P,. = 0:54 :{(P
a = f+ig; b = f+g
where i = /(-1) and f and g are the images for which the cross power
Ave”
x
x
0.54 *
x
x t
+ x x
x
x
da Reo ix
&
bu Luin ar ES” Hin
x x
fiisgetaooféofti potter 3 :
x 8 doin x ©
ay x % 1 wx .
$251 3*3 $ E ous
0 * $. À } + 1| ti :
T T T T T T T T T T T T T T T T H
] 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
(9).:49) 248): 49). (8) ::(9) 549) 49 HEY. {TY 0 (431 (2) 2 02) (4) :(0); (0)
Figure 5. Relative energies of the largest spike in phase cor-
relation function. Noncyclically displaced images.
Numbers within paranthesis indicate the number of
successful U-estimates out of 9.
spectrum is to be determined. According to Ulrych and Bishop (3),
estimates of power spectra, which are consistent with the fact that
nothing is known except the specified series, are obtained by
maximizing the entropy. This amounts to the same thing as determining
autoregressive models of the series from which the power spectrum
estimates are determined using the prediction error coefficients is
2 M 2142
P = 26 ] || *. E Y. exp(-i270v3)l (41)
ff jet J
where co^ is the variance of the error series, v is the frequency and M
is the order of the autoregressive process. As d, in (39) also can be
obtained from the cross power spectrum (40), one-dimensional tests
using synthetic image projections have been performed. The results are
promising when M is large enough. However, no results for real image
pairs (including noise and effects of different projection) were
obtained before the release of this paper.
DISCUSSION
The assertion of Kuglin and Hines, and also the results obtained here
indicate that the phase correlation function is so insensitive to
wrap-around effects, that it can be used also for large translations
u. Unfortunately, the phase correlation function lacks theoretical
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