International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part BI. Istanbul 2004
Figure 3a illustrates that a systematic trend is evident in the
residual Easting (across-track) direction,
especially in the surroundings of 1300m high Mt. Wellington,
whereas this error signal is substantially reduced when the
height correction is undertaken (Figure 3b). This supports the
view that projection discrepancies caused by the height
undulation are largely removed by the image conversion.
However, even with the height-correction the RMS values are
still marginally higher in Easting than in Northing. One of the
reasons for this is attributed to errors in the image conversion
arising from uncertainty in the value of the roll angle of the
sensor utilised for the computation of the height-corrected y-
coordinate. However, the accuracy achieved is still considered
to be impressive from a practical point of view. In fact, it is
noteworthy that sub-pixel accuracy was attained with the
standard formulation, Equation 1 (ie. without the height
correction).
vectors in the
4.3 QuickBird results
As mentioned in Section 3.3, it is expected that QuickBird
imagery will be more prone to adverse effects from
dynamically changing sensor orientation. To first verify the
presence of an error signal from the sensor dynamics, and to
quantify its magnitude in the imagery, spatial resections were
computed for three formulations of the affine model: 1) the
standard 8-parameter affine, 2) the extended affine with time-
variant parameters, and 3) the extended model with additional
parameters. The resulting RMS values of image coordinate
residuals are listed in Table 3.
It is apparent from the results in Table 3 that non-linear error
influences are present and that these cannot be fully
compensated via a standard aftine formulation, Equation 1. In
fact, the implementation with time-variant affine parameters
and additional parameters is required to yield satisfactory
results. It can also be seen that the magnitude of the non-
linearity induced image coordinate errors is case-dependent,
ranging here from roughly 5 to 16 pixels.
affine model, Equation 7. Similar, though much smaller
systematic error trends have also been seen in RPC bundle
adjustments of QuickBird stereo imagery (Fraser & Hanley,
2004).
Std. errors at
Number rc RMS discrepancies
of GCPs RMS, checkpoints (m) at checkpoints (m)
(CP) (pixels) : T
Ox / ON / Oy Sp Sy / Sy
QuickBird Melbourne stereo
Al (-).9 0.13 -/-/- 0.22 / 0.24 / 0.3
15 (66) 0.16 0.16 /0.15/ 0.24 0.26 / 0.29 / 0.42
12 (69) 0.16 017 ^0.157 0:26 0.257 0.32 / 0.41
10 (71) 0.16 0.18/0.15/0.26 0.327/0.36 / 0.43
QuickBird Yokosuka stereo
Al (-)'? 0.17 mi 0.37 / 0.40 / 0.48
15 (46) 0.20 0.17/0.217.0.36 0.50 / 0.48 / 0.60
12 (49) 0.19 0.19 / 0.23 / 0.39 0.49 / 0.47 / 0.56
10 (51) 0.19 0.21 / 0.26 / 0.43 0.49 / 0.48 / 0.54
*1). All GCPs loosely weighted (= 2 m)
Table 4. Results of affine bundle adjustments of QuickBird
Basic imagery of Melbourne and Yokosuka testfields.
X Y Y
\ X AA. + \
Y VN 1
, he à va 1
ASN i m
«X y X
N Ad \ t \ X 4
x WN \ 3 N A * s
A x % Ts À N ‘ \
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A Y
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Tos pixei 3
Jos pixel NN CA
RMS of image
Formulation residuals (pixels)
Left Right
QuickBird Melbourne
Standard 15.26 5.38
Time-variant parameters 7.90 2.48
Time-variant parameters plus AP 0.37 0.22
QuickBird Yokosuka
Standard 10.40 15.95
Time-variant parameters 1.63 8.27
Time-variant parameters plus AP 0.46 0.60
Table 3. Resection residuals from the affine model.
In light of the results in Table 3, the extended formulation of
the affine model with additional parameters was adopted for the
bundle adjustments of the QuickBird Basic stereo pairs. The
results of these adjustments are listed in Table 4. Figure 4
shows plots of the residual image coordinate error vectors from
the bundle adjustment for the all-GCPs case (the IC row of
results in each testfield). The systematic trends exhibited in the
plots of residuals, which show approximate alignment with the
satellite track direction, suggest the existence of higher-order
sensor perturbations, which cannot be modelled by the extended
Left stereo Right stereo
a) Melbourne
e
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queque evt
x A XN s Ni à À
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Left stereo Right stereo
b) Yokosuka
Plots of image coordinate residuals from affine
bundle adjustment for QuickBird (case of all GCPs).
Figure 4.
5. CONCLUSION
. This paper has discussed three formulations of the affine model
for HRSI sensor orientation, and summarized results obtained
with this empirical approach for geopositioning. In the context
