4. RESULTS 
Experimental results show, that the models discussed above 
have significant differences. Different combinations of fixed 
parameters and different combinations of integrated 
observations occur to be distinguished. In the following sections 
the differences are shown. Beginning with the most general case 
including as many observations and as many parameters to be 
estimated as possible, the number of estimated parameters and 
observations are stepwise reduced. 
4.1 Reproducibility 
To evaluate the usefulness of a certain model, the concept of 
reproducibility of the estimated parameters is used. The 
reproducibility is measured by the standard deviation of the 
results of a certain model using distinct sets of observations. 
While evaluating the reproducibility it has to be taken into 
account that the calibration parameters may be disturbed by an 
interruption of a single session, if an user takes off the glasses 
and puts them on again. In a session, the relative position and 
orientation of the user's head (measured by the tracker) and the 
glasses should not change. For this reason, the reproducibility 
can only be checked by a random sampling of a larger set of 
observations of the same session. 
4.) Number of estimated parameters 
First of all, it is searched the best constellation of the estimated 
parameters. All available observations are used and their 
residuals are estimated. The unknown parameters describe the 
transformation from the sensor co-ordinate system to the 
display co-ordinate system. Where: 
C; ,C7- focal length in row and column direction of the 
image co-ordinate system in pixel, 
Xo, o = principal point in pixel, 
X0 Yo Zo = components of the translation from the 
sensor system to the eye system in meter, 
O, ¢, K = components of the translation from the 
sensor system to the eye system in radiant. 
  
  
  
  
Par. Results 
absolute standard deviation standard deviation 
value (A) (A) (B) 
€; 1218.05 159.68 32.4118 
C2 1054.42 140.58 6.8377 
Xo 215.09 34.33 0 
Vo 236.91 37.40 0 
Xp 0.0347 0.0130 0.0068 
Yo 0.0325 0.0257 0.0035 
Zp 0.0537 0.0571 0.0085 
e 0.7782 0.3574 0.0764 
9 1.6918 0.0298 0.0077 
K 1.0981 0.2929 0.0813 
  
  
Table 1. (A): all observations used and 9 parameters estimated. 
(B): all observations used and constant principal point. 
In table 1 an example for a calibration result is given for two 
cases in which all observations are used. The reproducibility is 
rather weak for case (A). The co-relations between the 
parameters reaches up to 99%. That means, there are too many 
unknown parameters introduced into the model. Several 
experiments showed that the best strategy here is to keep the 
principle point fixed at the centre of the image. It can be 
observed from the values of the fourth column of table 1, that 
there is a significant increase in reproducibility. 
4.3 Number of observations 
In contrast to the section 4.2 the number of observations is 
decreased in the following. Firstly, the second bird is left away, 
secondly, the first bird is assumed to be error free. Then, using 
both birds, the number of image points is reduced consequently. 
In table 2 can be seen that there are significant differences 
between the conventional methods and the method proposed in 
this paper. Comparing column four in table 2 with column four 
in table 1 it can be observed that there is a decrease of accuracy 
when assuming the tracker to be error free (B). This is the 
assumption made implicitly when only image points are 
assumed to be affected by errors. The influence of the tracker 
errors can also be seen in column three of table 2. When 
assuming only the pixel co-ordinates to be error free and only 
the tracker errors are considered (A) the results are still better 
than (B). 
The last table 3 gives an impression how much points necessary 
to get reliable values. In the case of using only ten points, the 
focal length has to be assumed to be constant. Otherwise, there 
will occur cases that have no convergent solution when using 
the Newton method. The table 3 shows that the accuracy of the 
focal length decreases faster than the accuracy of the parameters 
of the transformation form the eye-system to the display- 
system. 
  
  
Par. Results 
absolute standard standard 
value deviation (A) deviation (B) 
€; 1388.9982 71.8874 98.2841 
C2 1359.0145 54.5555 77.1293 
Xo 215.09 0 0 
Yo 236.91 0 0 
Xo 0.0364 0.0074 0.0053 
Yo 0.0487 0.0077 0.0107 
Zo 0.0449 0.0443 0.0482 
1.9015 0.1411 0.0679 
Q 1.6715 0.0150 0.0221 
K 1.7994 0.1462 0.0632 
  
  
  
  
Table 2. (A): second bird left away. (B): the tracker is assumed 
to be error free. 
  
  
  
  
  
Param. standard deviations 
obs. num. 10 20 30 
cy 0 103.1610 32.4118 
€? 0 79.1103 6.8377 
XQ 0 0 0 
yo 0 0 0 
X» 0.0059 0.0046 0.0068 
Yo 0.0110 0.0093 0.0035 
Zo 0.0127 0.0495 0.0085 
œ 0.1127 0.0787 0.0764 
0 0.0142 0.0098 0.0077 
K 0.1160 0.0765 0.0813 
  
Table 3. Examples using different numbers of observed image 
points. A zero standard deviation indicates a fixed parameter. 
—538— 
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