2 
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the radial lens 
e asymmetric 
(8) 
nd substitute 
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ires principle. 
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e image set is 
imated values 
r the rotation 
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> points. The 
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ise in image 
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ee Figure 2). 
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he image set 
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cipal point 
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ir behaviour 
rameters are 
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ould expect, 
sets end the 
were totally 
  
  
overlapped by the others and because set 9 was already quite 
“massive”, image set 5 was chosen for later tests. 
3 
2 3 
  
  
| | 
| | 
E _ 
| 
LI 
  
  
  
  
  
  
  
  
  
Figure 2. Possible image sets. 
The affect of non-concentricity was studied by moving the 
projection center 1, 2, 5, 10 and 25 millimeters aside from the 
rotation center and repeating the simulations. Again Gaussian 
noise with 0.1 pixels standard deviation was added to the image 
observations. The results concerning the principal point 
coordinates and camera constant are shown in Table 3. They 
can be compared to the results shown in the first data row of 
Table 2 (same noise level). It can be seen that in this case a 
small deviation in concentricity has only a minor affect to the 
results. In the simulations the object point cloud was at the 
distance of 20-30 meters from the camera. If it would have been 
further, the less the non-concentricity would have affected. 
  
Noise std. px py € 
  
mean | std. | mean std. mean std. 
  
1 614.19 | 0.61 | 479.12 | 0.57 | 1404.05 | 1.33 
  
25 613.86 | 1.44 | 479.10 | 1.23 | 1404.31 | 3.24 
:3 614.16 | 3.20 | 479.09 | 2.95 | 1403.76 | 6.17 
  
  
  
  
  
  
  
  
1.0 612.89 | 6.06 | 478.77 | 6.16 | 1403.90 | 15.77 
  
  
  
  
set px py c 
  
mean std. mean std. mean std. 
  
2 | 614.61 | 7.68 | 485.18 | 44.20 | 1431.17 | 45.11 
  
3 | 613.81 | 5.35 | 480.89 | 30.66 | 1426.99 | 28.32 
  
4b | 614.26 | 4.60 | 479.43 3.04 1403.31 7.89 
  
5 | 614.16 | 3.20 | 479.09 2.95 1403.76 6.17 
  
9 | 614.24 | 2.03 | 478.97 2.03 1404.05 3.70 
  
  
  
  
  
  
  
  
| 140420 
  
| 614.12 | | 478.98 | 
  
  
Table 1. Results of the different image sets. The image sets 4b, 
5 and 9 are clearly better than the first two ones. On the last row 
are the correct values. 
The chosen image set was tested for noise sensitivity. Again 
synthetic images were created and Gaussian noise with four 
different standard deviations was added to the image 
coordinates of the corresponding points. The chosen standard 
deviations were 0.10, 0.25, 0.50 and 1.0 pixels, and the 
maximum noise elements were 0.35, 1.05, 1.91 and 3.5 pixels, 
respectively. In Table 2 are shown the mean values and the 
standard deviations of the obtained principal point coordinates 
and the camera constant. The obtained mean values comply 
with the correct values and the standard deviations of the 
parameters seem to be roughly linearly dependent on the noise 
level. 
  
[614.2] | 47898 | | 1404.20 | 
  
Table 2. The affect of the noise. The standard deviation of the 
parameters is roughly linearly dependent on the noise of the 
image observations. On the last row are the correct values. 
  
error px py c 
  
mean std. mean std. mean std. 
1 614.10 | 0.58 | 479.14 | 0.60 | 1404.11 | 1.26 
2 613.91 | 0.63 | 479.36 | 0.59 | 1404.00 | 1.35 
5 613.82 | 0,75 | 479.69 | 0.70} 1403.75 | 1.31 
10 613.55 | 0.80 | 480.59 | 0.66 | 1403.48 | 1.39 
25 612.72 1 1.07 | 48313 | 0.91] 1402602 12.12 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
V e| — amo] | 1404.20 
  
Table 3. The affect of non-concentricity. In this case the 
deviation can be more than one centimeter before it has some 
meaning. On the last row are the correct values. 
4. TESTS WITH REAL IMAGES 
In the following, some tests with real images were done. The 
images were taken with Olympus Camedia C-1400L digital 
camera. One irritating feature in this camera was that the 
automatic focusing couldn't be turned off. To avoid the situation 
where all the images have different focus the camera was 
always directed to a same point for focusing before image 
capturing. This procedure assures that images in one set have 
roughly the same focus, but between image sets there can be 
differences. 
At first, a careful test field calibration was performed. A bundle 
block adjustment software with the distortion model mentioned 
in paragraph 2 was implemented. About 400 image points were 
measured manually and the calculations were carried out. The 
—587—