ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
where Vv pt Ya” residuals at image coordinates and 
additional unknowns (object point coordinates (index 
p) exterior orientations (index f), additional self- 
calibration parameters (index a)) 
4, ,A,, 4,7 design matrices 
I = identity matrix 
X 3 ,X,,X,- change in estimated unknowns 
Ld 
polity = reduced observation vectors. 
Using an appropriate stochastic model e.g. weights P, for the 
different unknown parameters (e.g. EO parameters), this model 
covers the whole range of photogrammetric applications from 
direct georeferencing, where GPS/inertial data are used as fixed 
parameters (high weight FP, — o0) and object points are 
obtained from over determined forward intersection, to standard 
aerial triangulation, where the EO parameters are estimated as 
unknown parameters ( P, — 0) based on ground control points 
only. 
2.2 Influence of self-calibration parameters 
Within traditional aerial triangulation the use of additional 
parameters for self-calibration is broadly accepted. Using these 
additional parameters the physical process of image formation 
is adopted to the assumed mathematical model of central 
perspective represented with the collinearity equation. In other 
words, the additional parameters compensate for any remaining 
systematic inconsistencies between mathematical model and 
physical reality. Empirical investigations from Nilsen (2001) 
have shown average systematic image deformations around 5- 
10um for typical airborne photogrammetry projects. In 
especially when using direct georeferencing based on 
GPS/inertial only these systematic effects are critical since they 
remain unknown and will deteriorate the obtained object point 
accuracy significantly. 
Using additional parameters there are two different approaches 
for modelling: In the first approach physical relevant 
parameters like focal length and principle point corrections plus 
different types of image deformations, like radial, decentering 
and in-plane distortions are estimated. Such parameter sets as 
proposed by Brown (1971) are typically used for close-range 
camera calibration and implemented in commercial close-range 
photogrammetry packages (e.g. Fraser, 1997). On the other 
hand, self-calibration in standard aerial triangulation often relies 
on mathematical polynomial approaches as proposed e.g. by 
Ebner (1976) and Grün (1978). In contrary to the physical 
relevant parameters such polynomials are modelling in-plane 
distortions only, based on the assumption that other effects are 
negligible due to the strong interior geometry of standard 
airborne cameras. Furthermore, in standard airborne flight 
configurations variations in the camera interior orientation 
parameters cannot be estimated as far as no additional 
observations for the camera stations provided by GPS or 
imagery from different flying heights resulting in different 
image scales are available. The Ebner or Grün polynomial 
corrections are formulated as orthogonal to each other and with 
respect to the exterior orientation elements of imagery. This is 
of particular interest in case of GPS/inertial system calibration 
due to the strong correlations of GPS/inertial position and 
boresight alignment offsets to be calibrated with the exterior 
orientation of imaging sensor. Normally, the two modelling 
approaches are seen in competition, nonetheless the estimation 
of physical significant parameters and polynomial coefficients 
is supplementary and both models can also be used 
simultaneously, as already pointed out in Brown (1976). 
The influence of different additional parameter sets during 
GPS/inertial-camera calibration is illustrated with the following 
example from a real flight test. Within this test an integrated 
GPS/inertial-AT for system calibration based on ground control 
points and GPS/inertial measurements was performed, where 
the position offsets and boresight angles were estimated as 
unknown parameters in addition to additional self-calibration 
terms using the Ebner and Brown parameter model, 
respectively. The test data were taken from the calibration block 
presented in more detail in Section 3. The total influence of the 
estimated significant self-calibration parameters on image 
deformations is depicted in Figures 1 and 2. The unit vector is 
about 10pm and given in the upper left corner of the plots. 
CET TT Tay LL EL TETE 
150 —> 150 —> 
10 mu 4 10 mu 
. 4 ‘ *, 
100 P N 100 
F : E 
so - >» 4 \ sol 
cba baa ied 
»i99F $ > oe " > 
  
4 
3 
1 
50r 4 . , 4 d - . it «e 
100} , / -100 zt / / / 
E e H . J e 4 4 
150 
  
o 
alana laa 
A el eat lets 
Crutussudosusthsssstisustsos 
-150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 
X X 
Figure 1. Total deformations Figure 2. Total deformations 
(Ebner parameters). (Brown parameters). 
As it can be seen from the distortion vector plots in Figures 1 
and 2 the estimated total image deformation is significantly 
different, where the obtained O9 from AT is identical. Using 
Ebner polynomial coefficients the parameters model a sort of 
barrel-shaped distortions. The image deformation is about 
10pm maximum and therefore within the expected average 
range. In contrary to this, the deformations from the Brown 
parameter sets seem to be very irregular and significantly 
larger. They reach maximum distortions about 60pm. Since 
such large image deformations are highly unlikely for airborne 
frame cameras the Brown parameters over-compensate for other 
remaining systematic errors from object space. Due to certain 
correlations with the exterior orientation and boresight 
calibration parameters the physical interpretability of estimated 
Brown parameters is questionable. Self-calibration here partly 
compensates effects from the boresight parameters. In Table 1 
the estimated calibration terms are explicitly given dependent 
on the chosen additional parameter set. The estimated 
calibration parameters AZ, and A¢ differ significantly, 
resulting in different object point coordinates when applying 
this system calibration for direct georeferencing. The choice of 
appropriate self-calibration models to estimate true physical 
parameters is of major importance during overall system 
calibration. 
  
  
  
  
  
  
  
  
  
  
  
Calibration term Ebner Brown 
AX, [m] -0.110 -0.100 
AY [m] eliminated eliminated 
AZo [m] -0.262 -0.343 
AQ [gon] 0.4573 0.4598 
AQ [gon] 0.0668 0.0538 
AK [gon] -0.2897 -0.2907 
  
Table 1. Estimated system calibration parameters.