International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
  
  
  
  
  
  
  
  
  
  
  
Figure 3. Object coordinate (X, Y, Z), auxiliary coordinate 
(X', Y', Z) and Linear Array (0, y, z) coordinate systems. 
There are many systematic errors disturbing the ideal sensor 
model. The most important ones, with a distinct physical 
meaning, are: 
1) Radial lens distortion (2 parameters) 
2) Shift of principal point (1 parameter) 
3) Camera constant (1 parameter) 
4) Tilt and inclination of the Linear Array with respect to the 
rotation axis (2 parameters) 
5) Eccentricity of the projection center from the origin of the 
auxiliary coordinate system (2 parameters) 
6) Resolution of rotation (1 parameter) 
7) Tumbling (3 or 6 parameters) 
The above errors are modeled as additional parameters for a 
prototype of a panoramic camera. The results of the modeling 
for two different cameras were reported in Amiri Parian and 
Gruen (2003, 2004). 
The additional parameters can be divided in four different 
groups. The first is related to the camera head and optics 
(parameters of classes 1, 2 and 3). The second group of 
parameters (Figure 4) is related to the configuration of the 
camera head and the plane of the turntable (parameters of 
classes 4 and 5). The third group is related to the turntable itself 
(parameter of class 6). And finally the fourth group is related to 
the mechanical errors of the turntable, tumbling, while the 
camera rotates (parameters of class 7). Tumbling is mainly 
caused by an incomplete shape of ball bearings and the 
Figure 5. Effect of tumbling: Oscillation of the origin of the 
auxiliary coordinate system. 
contacting surfaces (Matthias, 1961). Tumbling results from the 
mechanical properties of the instrument. Especially, it is 
affected by the rotation around the vertical axis and shows its 
effect as a change of the exterior orientation of the camera head 
during rotation. From that, one of the main effects of the 
tumbling is the moving of the origin of the auxiliary coordinate 
system during rotation (Figure 5). For more detailed 
information on the mathematical modeling of the tumbling see 
Amiri Parian and Gruen (2004). 
In the next chapter we will report the physical measurement of 
the tumbling and the result of calibration and accuracy testing 
with/without the tumbling parameters. We show the results of 
accuracy tests with minimal number of control points. 
4. RESULTS 
4.1. Physical Measurement of the Tumbling 
The examination of the tumbling error of the SpheroCam was 
carried out by an inclinometer. In the present case the 
Zerotronic from Wyler Switzerland is used, which provides the 
inclination in a specific direction. The inclinometer was placed 
firmly on the top of the turntable near the gravity center of the 
camera system. Then using the operating software of the 
camera, the inclinations of at least 3 continuous rotations 
(10809) of the turntable at every 15° were recorded. To see 
  
Linear Array 
  
(a) 
  
  
  
#4 Turntable 
  
(b) (c) 
  
Figure 4. Additional parameters of the configuration of the camera on the turntable. (a) Eccentricity (ex, ey), (b) tilt of the linear 
array (/x), (c) inclination of the linear array with respect to the rotation axis (/z). 
26 
   
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