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4 1995 
  
  
351 
2.) Pseudo observations £; of 3D coordinate values of the control points with a diagonal cofactor matrix 
Qa | 
Lai + e, ML z (8) 
With these equations the tolerances of the calibration standard are modeled as stochastic variables. 
3.) Pseudo observations £p of all parameters p = [91/7 wife, Killer Mıle R3, K2;,, Pj, Po E, Qi/r] T with 
a diagonal cofactor matrix Q pip 
£p +e lp =D (9) 
These equations allow us to gradually turn on and off the estimation of the parameters. 
4.) Similarity constraints between the left and right image parameters in the stereo rig with a.diagonal matrix 
Qcc describing the degree of similarity. The degree of similarity will either be provided by manufacturing 
specifications or can be defined by the user. 
e:=B-p B contains the linear similarity functions (10) 
The solution of the Least Squares problem produces maximum likelihood estimations of p, (Y, x) and £/ as 
well as their cofactor matrices Qpp: Qoigi; Qzizi and Qyigi. Thus, especially the distortion terms, can be 
tested for significance. The cofactors of the unknowns can further be used to analyse the determinability of the 
distortion terms by computing their contribution to the trace of Qpp- A fast algorithm for this computation- 
ally expensive procedure is provided by the recursive Kalman Filter [Grün 1986]. Non-significant and weakly 
determinable distortion terms are excluded from the imaging model. 
5 RESULTS AND CONCLUSIONS 
The behaviour of the new calibration approach has been investigated with simulated data. To test the robustness 
in presence of noise, we perturb the simulated image coordinates and the 3D coordinates of the control points 
with Gaussian noise. Apart from Paragraph C below, all computations are based on a simulated control point 
distribution with a lateral extension of 800um and a lifting range of 200um. These values correspond to the 
best case available for the optics in the Zeiss Stemi 11. 
A. Determinability of distortion terms 
Distortion terms are only determinable when using multi-stereo views of the calibration standard. [Grün 1986] 
proposes to use the contribution of each distor- 
Orientation Contribution on tr(Qpp)[%] tion term on the trace of the cofactors Qpp as a 
Q[°] E Ky, Kj, Hh Ih, global measurement for its determinability. Con- 
£20. 8.04 3964 3223 022 0.17 tributions of more than 5% are intolerable. The 
=0,90 | 0.11 0.19 0.16 3.93 431 threshold of 5% can not be considered as a gen- 
= 0,90,180 | 0.10 0.09 0.14 415 4.46 eral rule. Dependent on the configuration and 
0,120,240 | 0.13 — 0.08 013 416 429 the errors of input data, other thresholds may be 
& 0,90, 190, 270. | 0.09 0:08 Q1 387 300 valid. A high contribution means that this dis- 
tortion term wrongly compensates for one or more 
parameters in the mapping function. A second view on the rotated grating increases the determinability of 
E, K, and Kz to tolerable values. Unfortunately, this action weakens the determinability of the decentering 
distortions. It is due to the high correlations between the decentering distortion terms and the additional 
translation vectors æi, which have to be estimated in case of multi-stereo views. These correlations can be 
partially overcome by using four views instead of only two or three. However, using even four stereo-views, the 
correlations between the translations and the decentering distortion terms remain in the range of 30%. These 
values have to be compared with the mean correlation of 7% between all distortion terms and the other mapping 
parameters. 
  
  
  
  
B. Robustness of the parameter estimations in presence of noise 
Robustness can be tested by successively increasing the perturbation of the synthetic measurements. As the 
true mapping and distortion parameters are known, the numerical behaviour of single estimations and groups of 
estimations can be studied by analyzing their errors. Some results of characteristic types of estimations are shown 
in Figure 6. The estimations in the third and fourth column never significantly exceed the propagated confidence 
intervals, which are centered with respect to their true parameter values. On the contrary, the parameters of the 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995