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349 
through identical optical elements. This will further increase the correlation between the left and right mapping 
function. Our calibration algorithm can fit the parameters for the left and the right images up to any user 
specified precision. 
The calibrated vision system has to provide control data for the moving robot in a stable reference frame. The 
relation between this reference frame Eg and the robots internal frames is calibrated at the beginning of a 
working session and can be recalibrated very quickly if its accuracy decreases. We choose the reference frame 
as following (ez, ey, e; are the base vectors of Eze, e4, those of the left micrograph frame): 
1.) ez; ^ eg, 7 sin(yi). This can be expressed by the condition x, — 0. 
2.) As described in Section 3, e, is parallel to the vertical motion of the motorized specimen table. 
3.) e, = e; Aes. 
This definition has the advantage that the mapping function is uniquely dependent on the microscope setup 
and not on the mounting of the calibration standard. Thus, sequences of calibration results can be used to 
investigate the temporal stability of the microscope setup. In addition, after a first calibration, good initial 
guesses for the various parameters are therefore available. In Section 3 we described the use of lithographic 
gratings as calibration standards. The coordinates of the crossing points relative to each neighbour are known 
with high accuracy. However, as the grating can be mounted arbitrarily, a further rigid body transformation 
x — To = ST from the grating frame Ej; into the reference frame Ez has to be estimated. The special choice 
of e, = e; (given through the vertical motion) simplifies S to So (which includes a single rotation (?) and zo 
to [zo, Yo, 0]. The weak perspective equations allow us to define xy arbitrarily. We choose xo = o. We can set 
the zero vector by the following procedure: A control point P on the grating which is imaged close to the image 
centers is defined as the principal point. Its image coordinates &;(P), £,.(P) establish the image references £oj, 
€,,- According to equation (2) and substituting both rigid body transformations which have been discussed 
so far, one obtains Zp = 0 and zp = 0. The 3D coordinate values of the further control points can then be 
derived relative to P. As will be shown in Section 5, some distortion terms are weakly determinable if exploiting 
only a single stereo-view of the calibration body. The problem can be solved with m stereo views of the rotated 
grating. As the rotation center is not equal to P, the lateral translations of P for the views 2...m are non zero 
vectors and thus have to be estimated. 
C. Adding distortion terms 
Image distortions are usually modeled with additional polynomials of the image coordinates. To the best of 
our knowledge, the distortion terms never have been derived for stereo light microscopy. In the CMO case we 
have to distinguish parazial distortions which are caused by defects of the zoom lens system and non-parazial 
distortion caused by the CMO lens. The latter typically exceeds 1 pixel. For low magnification the effects of 
CMO distortion are even observable by eye. Paraxial distortion of photographic lenses has been systematically 
investigated and successfully eliminated in close range photogrammetry. We expect similar behaviour for the 
paraxial optics in the zoom lenses. Thus, we use the well known approach: 
6:(Kı,Ka,Pı,Pı) = € (Kıp? + Kap‘) + Pı - (p* + 2€*) + 2Po£n 
6,(K1, Ka, Pi, Pa) = n-(Kıp? + Kap‘) +2PıEn + Pr - (p* +2n°) where PpP=£+1T (3 
The coefficients Kı,K2,Pı,Pz have to be calibrated independently for each zoom position. We will use only 
two discrete zoom positions for the robot motion control. Its repeatability is guaranteed by a motorized zoom 
drive with a very precise rotation controller. 
On the contrary, the model for CMO distortion has to be equal for all zoom positions as the CMO lens perturbs 
an intermediate image in front of the zoom lenses independent of their magnification power. In addition, the 
left and right images are distorted by the same CMO lens and therefore behave symmetrically. According to 
[Born and Wolf 1970], the third örder ray aberration and the fourth order wave aberration are connected by 
Q9 
——— and 69 x-—— , (4) 
Qu a 
where W(4 describes the perturbation function to the Gaussian reference sphere (Seidel aberrations). u, v are 
the normalized image coordinates at the exit pupil of the CMO lens 3. The Seidel aberrations are responsible 
for lack of sharpness and for changing the image positions (distortion). For positional measurements the latter 
must be eliminated by a calibration. The distortion term of ¥*) is given by 
m xX — 
o® = E+ 3°) (Eu + jv) , (5) 
  
3The normalization causes equation (4) to be independent of the magnification. 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995