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D. Conwezity of the Least Squares Objective Function ® 
The existence of any stable minimum can be checked by the convergence of the non-linear optimization. As the 
rotations are the most important parameters in our application, we stop the iterations if the maximum change 
of micrograph rotation is smaller than 10-°radians. It turned out that in nearly all runs the maximum change 
did not exceed 10~8radians after the fifth iteration step. An empirical verification that not any but the correct 
minimum of ® is found is given by the figures below. The noise perturbation for the image coordinates and the 
3D control point coordinates was fixed to a lo level of 4um and 0.5um for all test runs. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
(A1) (A2) (B1) (B2) 
mp -0.1815 — ont f-zz rr 0.021 
-0.04 + . E . na Y 
_ 006} teca see (A2) 1] 0.182 z oil! detail see (B2) i| — 0.0205 
s ke E Z -0.1825 - 8 oci bem rmm J B M 
3 SI 3 018} Lo eut 2 
= 5 = s s E I 
Z 9n = -0.1835 2 ool 2 00195 
. 2 © 
T ] -0.184 F 4 00 + J 0.019 
0.2 a À À 1 À A 0.1845 1 A A. À. 0 À À À 1 1 L 0.0185 r1 À À LL 
0 ! 2 3 4 5 6. 7 2 3 4 5 6 7 0 1 2 3 4 $ 6 7 2 3 4 5 6 7 
Numbers of iterations Numbers of iterations Numbers of iterations Numbers of iterations 
Figure 7: Sample of the convergence of p and w starting with various initial guesses. Both angles converge to their true 
value even if the first guess differs 6°. (A2) and (B2) are zooms of (A1) and (B1), respectively. They show that the parameters 
advance without any oscillations to the final estimations. 
We proposed a versatile algorithm for calibrating CMO microscopes in the environment of a micro-robot system. 
The algorithm is based on the Bundle Adjustment. The specifics of microscope calibration had to be considered. 
Firstly, we formulated weak perspective equations and a new distortion type was introduced. Secondly, we 
expanded the Least Squares objective function to minimize not only the image coordinate residuals but also 
errors of the calibration standard and contradictions between the left and right stereo mapping functions. We 
addressed the special problem of finding 3D calibration standards with comparable submicron tolerances for 
all dimensions. Finally we investigated the performance of the algorithm using simulated data. The powerful 
statistical tools allowed us to detect and eliminate some weaknesses in the estimation procedure. Future work 
will unite the calibration software with the hardware of the micro-robot system, including also the Laser- 
Interferometer. 
REFERENCES 
[Baltsavias 1991] E.P. Baltsavias. Multiphoto Geometrically Constrained Matching. PhD thesis, ETH Zürich, 
1991. 
[Born and Wolf 1970] M. Born and E. Wolf. Principles of Optics. Pergamon Press, 4 edition, 1970. 
[Grün 1986] A.W. Grün. Photogrammetrische Punktbestimmung mit der Bündelmethode, Mitteilungen Nr. 40. 
Institut für Geodäsie und Photogrammetrie, ETH Zürich, 1986. 
[Kim et al. 1990] N.H. Kim, A.C. Bovik, and S.J. Aggarwal. Shape Description of Biological Objects via Stereo 
Light Microscopy. IEEE Transactions on Systems, Man and Cybernetics, 20:475 — 489, 1990. 
[Richardson 1991] J. H. Richardson. Handbook for the Light Microscope. Noyes Publications, 1991. 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences’, Zurich, March 22-24 1995