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Table 1 Filters for Object Location 
  
  
a.) Filter 
N°] Name Optimization Criterium Filterfunction m. 
: T 3 -à - 
1 |Matched Filter SNR?zV(g *m)/V(n»m) D ET «29 z R wE *c 
2 |Wiener Filter for ó(z-2,) | RzE(g.,*m)/W(g *m) ; Eg (=) z EI M 
T wil P: 2 - 
3 |Inverse Filter for ó(z-E,)| E(e( rag Rog) z 2 
4 |Phase Correlation R(x) is) (R «BR, )~Y/2xg(=x) 2 Rzi/ 3 R 1/*.g7l 
gg 7 wee Gu 
  
  
  
  
  
  
+) also least squares, maximum likelihood, best linear unbiased estimate (BLUE) 
b.) Search function e(z) = m(x) « g,(z) with maz(e(x)) = £, 
— 3 = -— 
  
  
  
  
  
  
  
  
N°| Name Expectation E(e(z)) Autocovariance function R_(z) 
1 |Matched Filter BR +R *8(x-%.) RAR xR 
n 9g 2 n 1 
2 |Wiener Filter for $(z-2.) | (6-R7 «R,) *B(z-8j) R, i 
3 |Inverse Filter for §(z-Z,) S(x-#,) R +R 
i g . 84 
4 |Phase Correlation RI «Rr n (zi) 6 
Discussion: 
Y... The. first. filter m, = Rz wg (-@) is the most commonly usec matched filter (for a derivation 
see e.g. Castleman 1979, p. 210). In case n is white noise, i. e. RB, is a ô-function, the expec- 
tation of the search function e, is the autocovariance function Ra) of the template shifted 
by the unknown value £,. It is well known that this filter is also optimal in the least squares 
sense, where the difference f (g-9 ,)* dx between g and g; is minimized (Svedlow et al. 1976, 
McGillem and Svedlow 1977, Meyers and Franks 1980, Ryan et al. 1980). Furthermore m, also yields 
the best linear unbiased estimator (BLUE) for the unknown shift, i. e. it leads to the smallest 
variance in this class of estimators (McGillem and Svedlow 1977). Finally z; is also the maximum 
likelihood estimator if the noise can be assumed to be normally distributed (McGillem and Svedlow 
1976). The cited equivalencies are generally valid for least squares solutions (cf. e.g. Koch 1980) 
In view of these overpowering criteria there seems to be no chance to find better filters. 
Actually at least three others exist. The reason for their development was the experience that 
the matched filter in practice often leads to unreliable results namely to mismatches at very 
wrong positions. This is due to the local character of the optimization criteria. All three 
filters try to minimize the probability of a false match by sharpening the searchfunction. 
: - -1 
2. The second filter ma = (80H E eR explicitely tries to optimally separate between the 
true position £, and all others (Emmert and McGillem 1973). The main part of the search function 
is a 6-function. As g, can also be generated by convolution of the now unknown function ¢(x-Z 
with the given filter g(z) and subsequent degradation with n(x) (cf. fig. 1b) the filter m, 
actually is identical with the Wiener Filter for restoring the function $(z-E.). 
pa 
3. The third filter m, = a”! is the inverse filter neglecting the influence of the noise. The 
average search function is a ó-function located at the correct position. This filter seems to be 
superior to the preceeding. But generally this is not true. Though both filters Mo and m. are 
highpassfilters (for normal imagery) the transfer function M, (u) of m, is bounded if the signal 
to noise ratio VE, (u)/P, (u) is not going to infinity, which seems to be a realistic assumption, 
whereas M. (u) might have poles namely if G(u) has zeros.