International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004
subject and Heipke, 2002); both are shortly addressed in the
next paragraphs.
Just to roughly summarize, it is so far acknowledged that in
operational terms both approaches lead to accuracies more or
less comparable on the ground; one-step methods require less
ground control or, with less reliability, none at all.
There are still some open issues, though. From a theoretical
standpoint, the lack of information on the covariance matrix of
the IMU/GPS solution (usually not provided by the proprietary
processing software) still prevents to specify correctly the
stochastic model in both approaches; from a more operational
standpoint, how often should the calibration procedure be
repeated and what is the minimum reliable block configuration
to calibrate successfully, though several papers have been
published on the subject, has not yet being practically translated
in technical recommendations and an operational procedure.
2.2 Previous work on this topic
One of the major efforts up today towards clarifying the
performance of IMU/GPS systems has been the OEEPE test
“Integrated Sensor Orientation” (Heipke et al, 2002). Within the
test activities, the authors used a “two-step” calibration
procedure (Forlani, Pinto 2002) which modifies, as far as the
stochastic model is concerned, the method proposed in
(Skaloud, 1999). Later on, a l-step procedure has been also
implemented (Pinto, Forlani, 2002), looking for the minimum
reliable configuration for the calibration block, especially in
terms of number of GCP necessary.
In the following we present an improved 2-step method, based
on a more realistic stochastic model, again after a proposal by
(Skaloud, 2003). By applying all methods to the Pavia dataset
we try to find out whether this more complex model yields
benefits in terms of accuracy of the calibration compare to the
others; beside, we check how the performance of the various
approaches changes with changing calibration block
configuration.
2.3 The “two-steps” procedure
Let’s start from the collinearity equation:
rer + RÍsrf (1)
where: r= position of point i in object space, a cartesian
system L conveniently located in the block area;
r= image coordinates of point / in camera frame c;
r' RE, s; = EO parameters of image / (position of the
projection centre, rotation matrix from c
to L), scale factor for image point /.
Let assume the IMU/GPS positions be referred to the projection
centre and the IMU angles providing the rotation from IMU
(the body frame b) to L; r can be also obtained as:
Ei ug L bis a
r= tf mers + Ri” [SiR Hal (2)
where: Y IMUGPs = IMU/GPS-derived position of the projection
centre of image /, in the local frame L;
RJ = rotation matrix from body frame b to L frame;
R,°, a’ = calibration parameters: rotation matrix from c
to b frame; offset between the IMU/GPS-
derived and the photogrammetrically-
derived perspective centre position in the b
frame.
In the “two-step” procedure the EO elements obtained by a
standard ground-controlled bundle block adjustment are
compared with the EO elements measured by the IMU/GPS
data in the same flight. The calibration parameters (besides the
misalignment angles in the matrix R.’, an offset a° between the
IMU/GPS solution and the photogrammetric solution for the
camera projection centres was considered) are estimated as a
weighted average of the discrepancies between the EO of the
block adjustment and those provided by INS/GPS.
Since no information is available for the latter, only AT results
are used to get a weighted average of a, a, a, c, $, K; the
weights are derived from the standard deviations of the EO
parameters estimated in the AT. This should yield a more
consistent result, since whenever block geometry is weaker
(e.g. on the border strips) the EO elements, which may be
biased and poorly determined, will count less for the
determination of the calibration parameters. Correlations
between EO elements arising from block adjustment have been
neglected.
The effectiveness of this weighting procedure was reflected in
“OEEPE test” (1:5000 block) where RMS on check points
computed by direct georeferencing, were in the order of 7 cm
for N, E and 12 cm for elevations.
2.4 The “one-step” procedure
Comparing (1) and (2), the EO of image / is computed as:
HH = r'wugps + R o, ul
uh 3)
RI - RJ RP - Rd: RS RJ R^
|
where: R, = rotation from b to /j (the navigation system)
. measured by IMU;
Rye = rotation from / to the geocentric frame G;
RG" = rotation from object frame L to the G frame.
We therefore substitute r“mmuyors + Ra" a^ for rt and the product
AR A R, RP for R in the collinearity equations, removing
the dependence on EO parameters. The modified equations are
then linearized with respect to the components of the vector a
the angles o, ¢, K of RZ, the components of ors and finally
with respect to the ground coordinates of the tie points. The
functional model for the block adjustment is complemented by
the pseudo-observation equation of each IMU/GPS data (either
positions and angles) and the pseudo-observation equation of
the coordinates of the GCP. This is the base of the “one-step”
calibration procedure.
As far as the stochastic model is concerned, due to lack of
information (in terms of a variance-covariance matrix of the
solution) from the IMU/GPS data processing, often considered
a proprietary information, we assign positions and angles
accuracies according to manufacturer’s specifications, therefore
neglecting correlations arising from pre-processing.
2.5 An improved “two-steps” method
In a recent paper (Skaloud and Schaer, 2003) discuss the
correctness of the one-step and two-steps procedures. Since the
two methods are equivalent if the variance covariance matrices
are correctly propagated, Skaloud favours the two-steps because
its implementation is easier: at least the EO variances should be
part of the standard output of any bundle adjustment program,
while including IMU/GPS observations with time dependent
correlations in a one-step bundle software would be difficult.
The importance of properly accounting for the temporal
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