stima- 
à peak 
i filter 
imen- 
6) 
be able 
hen 
7) 
BS. 
repre- 
(not to 
inte- 
iance 
ting 
s that 
t in- 
itn a 
the 
2 is 
this 
ide 
terio- 
equen- 
t 2/0; 
for 
  
least squares variance. If we now specify C n and assume the noise to follow the AR-model then 
e is a Toeplitz matrix Com ^ íc = d expí(-o|z-j|), i.e. o is the correlation between adjacent 
noise elements. Then we can compare the variance ¢?. estimated from a correlation with white 
nr 
ratio V of the expected values of the variances then can be obtained from (the derivation is a 
bit tedious) 
noise (C=o% I) and the variance d?, from a correlation with coloured noise, especially C=c,, The 
Eat, CeC ) n-A 
MEE LAE a 
iCza?I) - 
E(g^ , |Czo 1) n-1 
where A(g J = gt c Com pt I / Ip Ot g_ in the case C= o? I lies within the range of the 
extreme eigenvalues DU > (1-p)/(2+p) and À enr = (1+p)/(1-0) of the Toeplitz matrix C rn for the 
number of pixels being large (n + e, cf. Grenander and Szegü 1958). The first term in eq. (18) 
results from the error in the normal equations; it is dominant to the second term (n-A)/(n-1) 
resulting from the error in the estimation of the noise variance. Taking o = 0.8, à value which 
has been proved to be realistic for multitemporal images (cf. Svedlov et. al. 1676), results in 
1/9 « A» < 9. This means that the standard deviation for the shift estimated from a least squares 
adjustment using the wrong covariance matric C = o^ I might be wrong up to a factor 3 in both 
directions. The standard deviation will be too optimistic if g_, the derivative of the signal, is 
long waved, i.e. without high frequencies. The estimated shift, however, is nearly not influenced 
by using a slightly wrong covariance matrix, especially it still is an unbiased estimator (cf, 
Koch 1980, p. 164). 
2. The effect of nonlinear filters usually is difficult to predict. The statistical properties 
of the median filter, though, especially with respect to contaminated edge images have been exten- 
sively investigated by Justusson (1981). His results can be used to predict the effect of the 
median filter on the restauration of the gradient which is necessary for point transfer (cf. sect. 
2.3.1) when applying the iterative least squares approach (eq. (10)), where aul and 95,4 have to 
be estimated from one of the two images. 
If one keeps the effect of a linear and a nonlinear filter onto the variance ge of the gradient 
of the noise constant one is able to compare their effect on the variance c2 of the signal gra- 
dient 9, and with eq. (13) onto the precision of the template matching. Geiselmann investigated 
the gain in precision based on the theoretical results by Justusson and on computer simulations 
(cf. Geiselmann 1983). Using the conventional way of calculation the gradient CN .,)/? the 
standard deviation c, is smaller by a factor 1.35 if the median with 3 or 3x3 pixels is used for 
ww 
restauration. If however the sharper gradient 9447797 is used the gain in precision is a factor 
1.6, showing the smoothing effect of the Sonvenciona! gradient computation. Moreover, if the 
median with 5 or 5x5 pixels is applied the gain is larger than a factor ë. It depends on the ratio 
of the height of the edge and the standard deviation of the noise. The gain is even higher, about 
20 % if the noise has a longtailed , e.g. a Student distribution. The findings are in full agree- 
ment with those from Yang and Huang (1981). They also showed that the effect of the median filter 
is greatly reduced if the edge is not sharp but a ramp. 
3.4 The sensitivity of the matched filter 
The effect of unmodelled geometric distortions in the correlation may cause systematic effects and 
reduce the total accuracy of the match. Specifically, we will investigate the influence of scale 
differences between the two images in concern. 
4. In the presence of geometric distortions there exists an optimal patch size for correlation 
(cf. Svedlow et.al. 1976) as small patches do not contain enough pixels and large patches cannot