3.3 The effect of filters onto the precision 
If another filter than the matched filter is used for object location the variance of the estima- 
ted shift will be larger. This might occur if one uses one of the filters for sharpening the peak 
of the correlation function or if the noise is actually coloured and one applies the matched filter 
assuming the white noise model. For simplicity we will restrict the discussion to the one-dimen- 
Sional case. 
1. The template and the signal are assumed to be passed through a filter 4, e.g. yielding g - 
h*g-Hg with E for the moment beeing the circulant matrix with kernel vector A cf. Rosenfeld, 
a 
Kak 1976). Assuming the noise to be white the least squares solution for £ from Ag. * D. = Gr 
with uncorrelated error equations is not optimal. It leads to the estimator 
^ 
me 
^ 
9e 
2 = 19" 9 g' Ag 
Jl vas = — (15) 
s (e! 4^ gg to PON 
Using the covariance matrix C TAG = Cz we obtain the variance of £ from error propagation 
2 =: 9H HT BE «fo! g^ gg )72 
= C n g {gl g./ (16) 
If we choose ^ such that #' = = e 2 ct I then eq. (16) reduces to eq. (13). In order to be able 
to use the exponential image model we have to transform eq. (16) into continuous form and then 
apply Parsevals identity (cf. McGillem and Svedlov 1976, Fürstner 1982). We obtain 
jUfWi RB, PL dh i 
‘ (47) 
4 
where P, is the power spectrum of the now continuous filter 5. We discuss two important cases. 
a. The commonly used transformation T.(g(z)) = dg(z)/dz yields the gradient image and may repre- 
sent edge images (cf. sect. 2.3). T. is a linear filter with transferfunction Æ(u) = -j2mu (not to 
be mixed up with the matrix Z above), thus with power spectrum P, = 4m%u?. Working out the inte- 
grals we obtain the variance of the shift ga. (g)J«z 9/4*02 «a3 /(417n). Compared to the variance 
gaz) 2 1/202 a^ /(41?n) , which one would reach with the optimal matched filter, correlating 
te gradient Signals leads to a standard deviation which is a factor 7.6 higher. This proves that 
sharpening the peak of the correlation function in order to improve the reliability does not in- 
crease the precision (cf. the experimental results by Anuta (1970)). 
b. An ideal low pass filter with upper frequency 4, Can be used to approximate sampling with a 
pixelsize of Ax. z l/Zu.. According to Fórstner (1982) there exists an optimal frequency if the 
number of pixels and the signal to noise ratio is kept fixed, namely u 3.38 / a, where a is 
the parameter descibing the sharpness of the image. For aood aerial Photos with a - 200 yum this 
leads to an optimum pixel size of Am = 30 um, which seems to be realistic. On the other side 
if one keeps the length d = n Ax of the object fixed oversampling does not change,namely deterio- 
rate the precision as long as a good approximation for g_ is used (if necessary) and all frequen- 
sies u < u, are represented. As the powerspectrum P, (u) of the gradient has its maximum at 2/a, 
t 
H 
one should at least use the band around this frequency (cf. Helava 1976), which is 10 Ip/mm for 
good imagery. 
The influence of the median filter is dicussed below. 
C. Eq. (16) can also be used to get an idea how reliable the estimated variance c2 is if we in- 
correctly assume the noise to be white when in reality it is coloured.For this we substitute 2? = 
by the inverse C^! of the covariance matrix actually used in the estimation process. We can now 
Study the effect of different covariances C onto the variance c2. The choice 7 = C o yields the 
   
      
          
   
   
   
   
   
   
     
     
    
    
      
   
  
  
   
      
   
     
   
   
    
  
    
       
     
  
  
   
    
—— ow o cy (D 
a (r^ n 9 wu € N - Uu - fX £(» 
— F9 l'en 
tn