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ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision", Graz, 2002 
  
(CFL). Also, the orientation angles were fixed during this 
adjustment. This will require adjusting other parameters also 
to produce the balanced curve in addition to radial distortion 
parameters kl, k2 and k3. The other parameters are principal 
point shift, decentering and affine distortion parameters. Since 
these parameters are not independent of the radial distortion 
parameters, the equalization procedure will be repeated until 
we get the balanced curve. All of that processing was handled 
automatically except for the red camera was balanced 
manually. Figures 7 and 8 show the scaled magnitudes of the 
radial distortions and their orientations throughout the image 
plane with respect to the principal point (PPS) and the fiducial 
center (FC) of the image. 
= du Beta 
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in ONU déerubon 
= atts Harn 
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p 
  
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Figure 5. radial distortion for the Blue camera 
Exposure station 
  
  
  
Figure 6. image plane cross section 
5.2 Decentering Distortion 
When a lens is manufactured, all its components should be 
aligned perfectly. But such perfection is not possible. The 
misalignment will lead to systematic image displacement 
errors. This undesired geometric displacement in the image is 
called decentering distortion. In this calibration procedure the 
following mathematical model is used (Samtaney, 1999; 
Atkinson, 1996). 
6, 2 p[r! *2x-x Y] 2p, x, Xy y.) 
ó, 2 p,[r? *2(y - y, ]* 2p (x - x, - y.) 
(7) 
This was given earlier in equations 2 and 3. As mentioned 
earlier, the equalization procedure for the radial distortion has 
an affect on the other parameters. So, in each equalization 
iteration, the decentering parameters will have new values 
since their behavior will be adjusted according to the 
modification of the focal length. Nevertheless, the equalization 
technique does not change the final corrected coordinate values 
and the main purpose for it is to make the distortion correction 
balanced in magnitude. 
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8 * npa 
d = 
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ue a s 
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4 d z «1 ü 2 + 4 
x axis in (maj 
Figure 7. Scaled Radial Distortion on image plane 
centered at PPS for the Blue camera 
y axis n mimi 
P is 
£ E 
¥ Fin dn (min) 
Figure 8. Scaled Radial Distortion on image plane 
centered at FC for the Blue camera 
6. RESULTS AND DISCUSSION 
The resulting calibration parameters for the four cameras are 
summarized in the table below. Those parameters can be used 
to refine the coordinate observations in image space for each 
camera, respectively. We tried during this work to automate