4, The covariance functions P, and R, show that both filters prefer one of the both functions 
go c 
1 
g Or g.. The fourth filter m,z(E *R, J'Y :«g(—m)- (m,*m,)!/? is a compromise namely the geometric 
£ 4 : 
| SF} 
7/9 
y = 1 
= (BR «RO )U/x*g(-z)*g.(x) is Fully symme- 
€ v 
} 
/ 
mean of the filters m, and m.. The séarch function e,(z 
tric with respect to the given functions ¢ and g.. The covariance function 2, is a ó-function. 
c 
The Fouriertransform C,(u) - G* G. / |G* G,| of c, contains only the phase of the crosspower 
— 
7 - 
spectrum G* G. of ç and g,. This filter m, is therefore called the phase correlation filter 
(Kuglin,Hines,1975;Pearson,Hines,Golosman,Kuglin, 1977). The formulas given by Pratt already 1974 
are a discrete version of c,(z) and are motivied by prewhitening both functions g = R7!/?4g, 2,7 
TD 
= 
Fas 
je pa 
;/i«g, and then correlating g and g,. The phase correlation method thus treats all frequencies 
of the crosspower spectrum with the same weight leading to results which are very robust to narrow 
a 
J 
banded distortions or long waved disturbancies. The pictures given by Oppenheim (1981) demonstrate 
that the phase of the amplitude spectrum really contains the main part of the geometric information 
of an image. 
5. A comparison of the autocorrelation functions R, of the search function e(z) reveals that in 
case the signal to noise ratio is large the three filters Mos My and m, lead to similar results; 
in the noisefree case they are identical. On the other side if the noise component in g, is large 
one needs a good estimate or a priori knowledge about the covariance function Ans except when 
using the phase correlation technique, which does not need any a priory information about z. The 
matched filter m4 obviously is most sensitive to assumptions concerning the noise as, in contrary 
to m, and mz, errors in R, and thus in RB. do not compensate but cumulate. This is the price to be 
é & 
payed for obtaining the locally best solution. In case R, and R, are proportional, i. e. R -const. 
: J 
+ 
m all four filters are identical. This situation realistically can be assumed if images are con- 
taminated not only by film grain noise but also by long waved distortions, either geometric or 
time dependent ones (cf. Emmert and McGillem 1973, Svedlow and McGillem 1976), but also is met in 
radar signals where this type of filter is well known (cf. Urkowitz, 1953). 
6. The filters can also be used in a slightly modified form when in addition to the noise and 
the shift the template is passed through a linear filter A(z) before being observed, which might 
represent smearing effects caused by the athmospheric turbulence or the movement of the sensor. 
Then g, = hagxô(x-2,)+n and thus Bg, = R,0R +R, with Æ,=h(z)xh(-=). Then all filters have to be 
modified by substituting g by gxh, e. g. now the second filter reads as (R_ xR. +R) ‘wg(—z)*h (=) 
15718, Á w/o 
As the filter h(x) usually is not known precisely on uses a symmetric surrogate A(=)=h(—). But 
then 2, only is unbiased if the true function A(z) also is symmetric. Otherwise, e. g. in case of 
onesided illumination effects, systematic errors have to be expected, as e. g. Efe, rc) pri 
1 
2,7] = Ox 
n 
E «n 5»hxó(z-$.) does not have its maximum value at zzZ., because 7 ^«^ is not symmetric with re- 
+ 
+ 
+ 
spect to z=0. The bias E($.)-Z. in addition to 7 depends on a. and &.. 
7. The optimality and the linearity of the filters m., m, and m, is only given for known correla- 
4 
tion function R, thus known Ra,» If R, or R,, is estimated from g, e.g. by cyclic correlation 
À 4 w [| 
=7 
or what is equivalent by taking the inverse Fourier transform of the empirical cross power spectrum 
> 
the analysis of the properties of the search function becomes much more involved as the filters are 
not linear anymore. This especially holds for the phase corrrelation technique the way it was ori- 
ginally formulated by Kuglin and Hines (1975). On the other side, Emmert and McGillem (1973) and 
Pratt (1974) proposed to approximate the autocovariance function, namely by assuming an autoregres- 
- : i Lim je 4 4 t . 15 . . . - 
sive image model (cf. section 3.1). A similar adaption leading to a nonlinear filter is the common 
standardization of the cross covariance function gí-z)wg.(z leading to the cross correlation 
function e(z)=g(-=)+a,(z)//R (Gj*R, (o) where an estimate 0%, TR, (0) is used for the variance 
"i me = 
= 
ba 
of the signal Z;- Cross correlation not only is invariant to unknown brightness differences of 
and g. but in its discrete and local version is highly adaptive to varying illumination. 
a 
z