AA css 
an EEE -—— 
  
Figure 1. The used rotation tool. The camera can be rotated both 
horizontally and vertically so, that the projection center stays in 
one point. 
2. MATHEMATICAL MODEL 
If two images are taken from the same point, the relationship 
between the corresponding image points is formulated as 
x X) 
Yı m sR Yr) (4) 
m6 ze 
In Equation 4 x;, y;, x; and y; are the ideal distortion free image 
coordinates having origin in the principal point, R is a rotation 
matrix and 
   
2 
2 (5) 
    
2 2 
xX +x +c 
2 2 
X) * X5 C 
Unfortunately the measured image coordinates are usually far 
from the ideal ones. There are two stages in the image capturing 
process which distort the image. First, the lens causes non- 
linear distortion and second, the camera's CCD-sensors 
geometry can deviate from the assumed geometry. Traditionally 
these distortions have been corrected simultaneously, but in this 
study they were done separately in two steps like suggested in 
(Niini, 2000). In the first step, a linear correction was applied to 
the measured image coordinates x, and yo. 
xp) 7*p + f(y -Yp) 
(6) 
yj * a(yg 7 yp) 
In Equation 6 x, and y, are the principal point coordinates, a 
compensates the scale difference between the two coordinate 
axes and / takes care of the skewness of the axes. This 
operation can be seen as a transformation from the image to the 
CCD-sensor. Now a non-linear correction is applied to these 
linearly corrected coordinates x, and y;, 
Xj = xj + XJ (yr? zd: kara + r9) == Pq T s: 2x7) + 
*2poxpy. 
(7) 
2 
Vi zy tyr yr! ek po 02 252), 
+ 2piXjyj- 
In Equation 7 k,, k and k; are the parameters of the radial lens 
distortion, p, and p; are the parameters of the asymmetric 
distortion and 
2 
r=\x Eu, (8) 
If we do the multiplication in Equation 4 and substitute 
Equations 6 and 7 in it, we get three equations which include 
measured image coordinates as observations and the rotation 
matrix elements and the camera parameters as unknowns. Each 
point pair gives three equations. Rotation matrix elements and 
camera parameters can be solved using least squares principle. 
It requires the linearization of the non-linear equations and 
some approximated values for the unknowns. If the image set is 
sufficient (enough images and overlap) the approximated values 
can be quite weak. The approximated values for the rotation 
matrix elements can be calculated using Equation 4 and the 
observed image coordinates of the corresponding points. The 
initial principal point can be set to the center point of the image 
and the camera constant can be set equal to the image width. All 
the distortion parameters (k;, k,, ks, p1, p», and f£) can be set to 
Zero except a which is set to 1. 
3. SIMULATIONS 
As mentioned before, three different subjects were studied by 
simulations; the affect of non-concentricity, requirements for 
the image set structure and the affect of the noise in image 
coordinate measurements. 
An infinite number of image sets can be created by varying the 
rotations, the overlap and the number of images (see Figure 2). 
During the simulations it was found that by having 50% overlap 
and a symmetric set structure, the calculations converged in 
most of the cases. Problems occurred only when the image set 
was very small or when none of the images was totally 
overlapped by the others (this might leave some part of the 
image without observations). In order to get some clue of the 
reliability of the obtained results, sets 2, 3, 4b, 5 and 9 (shown 
in Figure 2) were tested. For testing, synthetic images of size 
1024x1280 pixels were created based on certain camera and 
orientation parameters. A random object point cloud was 
projected to the images and Gaussian noise was added to the 
image point coordinates. Depending on the set there were 20- 
100 observed points on every image. One hundred simulation 
runs were done and the solved parameters were written to a file. 
The simulation results concerning the principal point 
coordinates and the camera constant are shown in Table 1. The 
other camera parameters are not listed here, but their behaviour 
was very similar. The mean values of the solved parameters are 
very close to the correct ones for the sets 4b, 5 and 9. The 
deviation is the smaller the more images, as one could expect, 
but there is a clear difference between the first two sets end the 
rest. Because none of the images of the set 4b were totally 
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