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.