The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008 
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X = X 0 +R-x 
where R = Rotation matrix 
x = Coordinate vector of an object point defined in 
the scanner or camera coordinate system (x,y,z) 
X = Coordinate vector of an object point defined in 
the superior coordinate system (X,Y,Z) 
X 0 = Translation vector between scanner or camera 
coordinate system and superior coordinate system 
2.1 Terrestrial laser scanner 
The geometric model of terrestrial laser scanners used for the 
following investigations was already presented in [Schneider, 
2007]. It bases on spherical coordinates (distance, horizontal 
and vertical angle) as observations (Figure 2, Equation 2). 
Figure 2. Geometric model of terrestrial laser scanners 
[Schneider, 2007] 
(1) AD = a 0 + a,D 
/"5\ 
Aa = 6, sec [} + h 2 tan p + [ô 3 sin a + b 4 cos a\ 
AJ3 = c 0 + [c, sin (i + c 2 cos p\ 
Cyclic distance errors were not considered as the used laser 
scanner is a time-of-flight scanner, where cyclic errors are 
supposed to be not existent. 
In comparison to the model described in (Lichti, 2007) two 
extra additional parameters were used in the geometric model, 
which could be determined significantly (Equation 4): 
• eccentricity between collimation axis and vertical axis b s 
• eccentricity between collimation axis and trunnion axis c 3 
A a, = arcsin— « — 
e D D ( 4 ) 
Aß = arcsin^- * 
D D 
Investigations of terrestrial laser scanners have shown that it is 
almost impossible to determine calibration values, which are 
effective and stable for the whole range and field-of-view as 
well as under different measurement conditions (e.g. Böhler & 
Marbs, 2004). In fact, correction values depend on a multitude 
of influences which can not be assumed to be invariable (object 
properties, area of distance, brightness, etc.). These 
circumstances can be considered by the implementation of a 
self-calibration procedure into the actual data processing. 
Moreover, observations are often already pre-corrected in the 
laser scanner instrument using correction models which are not 
published. This fact causes additional problems regarding the 
determination of significant and stable calibration parameters. 
D = t]x 2 + y 2 + z 2 
a = arctanl — 
x 
P = arctan 
2.2 Fisheye camera model 
The geometry of the projection of fisheye lenses does not 
comply with the central perspective geometry. At TU Dresden, 
Institute of Photogrammetry and Remote Sensing, a strict 
geometric model for fisheye lenses was therefore developed and 
successfully implemented, which allows for the calibration and 
precise orientation of a camera with fisheye lens (Schwalbe, 
2005). 
Equation 2 can be extended by correction terms AD, A a, Af in 
order to compensate for systematic deviations from the basic 
model, which allows for the calibration of the laser scanner. 
The following additional parameters according to (Lichti, 2007) 
were included into the investigations presented in this paper: 
. distance offset ao and scale a t 
. collimation and trunnion axis error and b 2 
. vertical circle index error c 0 
• horizontal and vertical circle eccentricity b 3 , b 4 and c/, c 2 
The appropriate correction model is defined as follows (Lichti, 
2007): 
Figure 3. Geometric model of fisheye lens cameras