330 
where $, à describe normalized and projected object coordinates at the entrance pupil of the CMO lens system. 
E is the coefficient of CMO distortion. The values $, ÿ can be approximated with good accuracy by £/M and 
n/M. 
In stereo vision, the image distortion causes a 3D object deformation relative to the intersection of the left and 
right chief rays. Therefore, Ö(£9;) and 8(&,,) can be set to o. After some algebra and considering that the chief 
rays intersect the CMO lens at the position &. = [+b,0]" *, the CMO distortion for the left image turns out as: 
8¢(E) = —Ep*(§ —b) + EEb(26 — 3b) and &,(E)- —E$(f -b(26—0)) , (6) 
where £ = (€ — &)/M, ÿ = (n — mo)/M and 0? = € +. 
Substituting b with —b yields the expression for the right image. Note, that the value of E is independent of 
the magnification. Thus, the values of 6 become smaller with an increasing image scale factor. Furthermore, it 
can be shown that the solution for E must be negative. 
   
  
  
  
2400 2400 
ë 1600 ë 1600 
= a 
E 800 E 800 
3 0 2 0 
= = 
© -800 = -800 
3 3 
? -1600 9? -1600 
>» > 
-2400 -2400 
-3200 -1600 0 1600 3200 -3200 -1600 0 1600 3200 
x-coordinate [microns] x-coordinate [microns] 
(A) (B) (C) 
Figure 5: Effect of CMO distortion in the left (A) and right (B) image. The grids with a point signature at the crossings 
are the deformed images of those grids without signatures. (C) shows the deformation of two parallel planes in presence of a 
non-zero CMO term E. Parallel, equidistant planes are deformed to non-equidistant curved surfaces. Note that the plane at 
level 0 is tangential to its correspondent curved surface. This is due to the 3D deformation relative to the intersection of the 
two chief rays, which is located at [0, 0, 0]7 . 
Further image deformations may originate from the frame-grabber. They can be compensated with the simple 
expressions §¢(a) = a - n and 6,(s) = s - n. The term a compensates for a shear in the image coordinate frame, 
s describes unequal scaling in £- and 7-direction. It turns out that s is weakly determinable in presence of E. 
A good initial guess can be computed outside the calibration procedure using the specifications of the frame- 
grabber and video cameras. However, some inaccuracies of the apriori s will be compensated by E, even if s is 
eliminated from the list of unknowns. This can be understood considering Figure 5 (A) and (B). A good part 
of E compensates for a relative scale distortion between £- and r-direction, as a non-zero s would do. 
  
  
D. The complete Gauss-Markov Model 
The mapping parameters as well as irregularities of the calibration gratings are estimated simultaneously with 
the non-linear Gauss-Markov Model. The model is generated by four types of observation equations. Input data 
consists of left and right image coordinate pairs of m stereo views on n target points. Obviously, n coordinate 
triplets of the 3D calibration standard have to be available as input data. 
1.) Image coordinate observations ? with a cofactor matrix Qe 6 
(Er Si) UV Tre Mir Bieten (Si d mi) -óg(Ki,,,, F2, Puy s Py, E, ayy) 
: k=} 
(7) 
i ={1---n},j = {1---m}; the power series of perspective distortion is expanded until CT DD 
diag(Qe 3 Therefore, the accuracy is not decreased by the usage of weak perspective equations instead 
of the full perspective imaging functions. 
  
  
4_p for the left image, +b for the right image; b is the distance between the intersection point and the optical axes of the CMO 
lens (see Figure 1) 
5The equations are based on the model for CMO light microscopes. In case of SEM, merely the parametrization of the distortion 
function 67 has to be exchanged 
6 For simplicity, the correlation between ¢ and 7 in the matching algorithm is neglected. Thus, Qee is a diagonal matrix. 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995 
  
eit nea a ee