For the estimation of the parameters ag and a, we consider the expression log e of 
equation (8): 
| M 
log ity 
I 
logy - log(l-y) * a4x + ag 
in our case 
i log i aj log SNR + ag (7) 
Fig.9 shows the approximated curves P(SNR) for all objective functions according to 
(7). Also their 95 % confidence interval and the measured probabilities p are plotted. 
The significance of the fits has been verified by chi-square-testing. 
Objective function classification 
To classify the correlation functions limiting values are extracted from fig.9. The para- 
meter Sg. gs. 89.50 and 89,95 denote the SNR witha probability of 5 76, 50 76 and 95 % 
respectively. Table 3 presents the estimated parameters with their upper 95 % confidence 
  
  
interval. 
Table 3: SNR values with P = 0.05. 0.5 and 0.95 
Objective 85 % 95 % 935 % 
function 50.05 confidence 30.50 confidence 30.95 corfidence 
a 0.14 0.24 0.34 0.50 0.82 1.40 
b 0.15 0.26 0.34 0.40 0.75 1.25 
C 0.42 0.78 1.40 2.20 4.80 12.00 
d 0.08 0.12 0.19 0.28 0.50 1.00 
  
The classification criteria listed in table 3 allow us to come to the following conclu- 
sions: 
(i) Best correlation probability is shown by the phase correlation method d). A SNR of 
1.5 or lower in our imges requires the application of this objective function to achieve 
most probable correlation. 
(ii) The objective functions a) and b) have almost the same probability of correct cor- 
relation showing no significant differences. So other criteria (e.g. precision or computa- 
tion time) may decide which one has to be used. If image SNR is above 1.5, functions 
a). b) and d) can be employed without preference. 
(iii) A poor probability is associated with function c), the Laplace coefficient. Wrong 
correlation even occurs at an image SNR of 9.0. Although a very fast function (BAILEY 
et al., 1978), it shows unsufficient probability and should therefore be handled with 
care in the correlation process. 
With these limiting values it is possible to control the application of objective functions 
in an automatic manner. If the SNR is known, it is possible to decide which correlation 
function can be chosen and to estimate the according a priori correlation probability. 
So the image SNR is all an automatic correlation controller has to know. But how can 
this be calculated if image and noise are almost inseparable ?