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should be viewed as a generalised form of the extended affine 
model. In some circumstances, constraints may need to be 
imposed on certain parameters to enhance model stability. For 
instance, if the object space system is locally shifted to the 
scene centre, the constant terms of the planes IT, and I, could 
become close to zero meaning that the parameters 4, and 4, 
may then need to be constrained as time-invariant. 
Equation 4 can be supplemented with additional parameters to 
describe quadratic or possibly higher-order error effects. These 
can be cast as functions of either image or object space 
coordinates. The choice of additional parameters depends upon 
the type of sensor. However, the first priority for the affine 
model would be four quadratic terms. In this case, the extended 
formulation is expressed as follows: 
Iz AXE AY 04,002 4,0) B4 B, (7) 
s=A;(OX +A ()Y + A, (1) Z + A(t) + B,I* + B,s* 
where  4;, B; = time-variant/invariant parameters 
1, s = line, sample coordinates 
X,Y,Z = object space coordinates 
Geometrically, the adoption of the additional parameters allows 
the image planes I'l,” and I1, to be curved. Further investigation 
will provide more insight into the connection between the affine 
and higher-order rational polynomial models. 
4. EXPERIMENTS 
4.1 Overview 
To verify the applicability of the affine model to HRSI, three 
sets of testfield HRSI data have been investigated. The aims of 
the experiments were, first, to ascertain the degree of accuracy 
degradation, if any, of the affine model when employed in 
mountainous terrain; and, second, to evaluate the extended form 
of the affine model (Equation 7) for QuickBird Basic imagery. 
Table 1 summarises the testfield data involved in the 
experiments. Further details of the testfields can be found in 
Fraser et al. (2002), Noguchi et al. (2004) and Fraser & 
Yamakawa (2004). 
  
  
  
  
  
  
  
Hobart Melbourne Yokosuka 
120 km? 300 km’ 300 km” 
Area = 
(1x10 (17.5x17.5) (17.3x17.5) 
Elevation 1280 m 50m 170 m 
No. of 110 points 81 points 61 points 
GCP/CPs by GPS by GPS by GPS 
Types of Mainly road Mainly road Mostly corner 
fare roundabouts roundabouts features 
point 
IKONOS Geo QB basic QB basic 
Image ; s : =, : 
= stereo pair stereo pair stereo pair 
coverage 
B/H = 0.8 B/H=1.0 B/H= 1.0 
Image Digital mono comparator 
Meas. (estimated accuracy of 0.2 pixel) 
  
Table 1. Summary of the Hobart, Melbourne and Yokosuka 
testfields. 
145 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004 
4.2 Results with IKONOS 
The results of affine bundle adjustments of the IKONOS Geo 
stereo pair imagery covering Hobart (captured in Reverse 
mode) are listed in Table 2. Because of the mountainous nature 
of the terrain, the standard affine formulation coupled with the 
iterative height correction approach (Subsection 3.2) was 
employed. To assess the effect of the projection discrepancies 
due to the influence of terrain, the standard formulation without 
the height correction was also applied (numbers in brackets in 
Table 2). It should be noted that several GCP configurations 
were tested for each GCP set (same number of control points), 
except for the case of all GCPs. The RMS discrepancies and 
standard errors shown in each row of Tables 2 and 4 are 
representative values for each GCP set and they are calculated 
from independent checkpoints only. The term “RMS,” in 
Tables 2 and 4 denotes the RMS value of image coordinate 
residuals from the bundle adjustment. 
  
GCP RMS. Std. errors at 
a checkpoints (m) 
(CP) pixels 
RMS discrepancies at 
checkpoints (m) 
  
Of / On / Oy Se / Sx / Sy 
AC) 0 913 m 0.54 (0.76) / 0.34 / 0.56 
9 (101) 0.13 — 0.49/0.46/0.85 0.62 (0.83) /0.43 / 0.73 
6 (104) 0.13 0.56 / 0.50 / 0.86 0.62 (0.84) / 0.41 / 0.72 
4 (106) 0.13 0.77/0.75/1.19 0.62 (0.90) / 0.42 / 0.77 
  
  
9 GCPs, none on Mt Wellington (101 checkpoints) 
9 (101) 0.13 0.70/0.65/1.06 — 0.70 (0.83)/0.46/ 1.10 
*1) All GCPs loosely weighted (0 3 m) 
  
Table 2. Results of affine bundle adjustments for IKONOS Geo 
stereo image pair covering the Hobart testfield. 
Overall, sub-pixel accuracy was achieved for all GCP sets, 
except for the case of 9 GCPs where none of the control points 
were selected from the Mt. Wellington area. The height 
correction produced an accuracy improvement in RMS 
geopositioning of about 0.2m in the Easting direction. This is 
because the image conversion for the height correction directly 
affects the sample coordinates, which correspond to the Easting 
direction in this case. To illustrate the effect of the height 
correction, the planimetric RMS discrepancy vectors from the 
all-GCPs bundle adjustment (the 1* row of Table 2) are plotted 
in Figure 3 for the cases of with and without height correction. 
  
  
  
a) Without height correction b) With height correction 
Figure 3. Plots of planimetric discrepancy vectors for the all- 
GCP bundle adjustment, Hobart IKONOS image.