age for each
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1976). Using
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iple point is
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Distortion is
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ased on the
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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
indieralare
in ONU déerubon
= atts Harn
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mius
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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3 \ / Fa
4 y
1 5n 4, 3 =
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& ü sl m—. € +FC "o afe fps
8 * npa
d =
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27 TAN
> Pa Ë 4 %
ue a s
m
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