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parameters 
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rom zero. 
(7) 
rough Chi- 
  
if a ME dg (a,df) accept H, 
otherwise, reject H, 
where: 
a is the chosen level of significance, and 
df is the degree of freedom, 1 in this case. 
If the null hypothesis is accepted, which means 
that the specific coefficient equals 0 and does not 
contribute to the form of the curve, the polynomial order 
is reduced by one. This test is carried out for each 
coefficient from higher to lower orders until the null 
hypothesis is rejected. 
4. Results with Simulated Data 
To evaluate the performance of the new model, 
many experiments were conducted using simulated data. 
We assumed a strip of four images containing one linear 
feature. The interior orientation parameters of the 
camera and the exterior orientation parameters of the 
images constituting the strip are pre-defined. Along 
these images both tie points and feature points were 
simulated. The image coordinates of all object points 
and features were computed using the known exterior 
orientation of each photograph. 
Experiments were conducted using the 
simulated image coordinates of the tie points and points 
along the linear feature as image observations (degraded 
by random errors of image coordinate measurement), 
and the exposure station coordinates and the linear 
ground feature as GPS measurements. We wanted to 
find out how the new technique behaves under different 
conditions, such as varying GPS accuracies at the 
perspective centers and on the ground. Its performance 
was evaluated based on the differences between the 
estimated ground coordinates of the tie points and their 
real values. In figures 1 to 4, these differences are 
displayed as error vectors originating from the true 
position of each point to the computed one in X and Y 
directions respectively. 
Figure 1 shows the results obtained by 
assuming a GPS accuracy of 1.0 m both on the ground 
and at the exposures stations. The mean error at the tie 
points after triangulation is about 0.5 m, which is quite 
acceptable considering the low accuracy of the GPS 
observations. A closer investigating of figure 1 reveals 
that these deviations correspond to a scale error. In other 
words, the accurate spacing between the exposures (base 
length) were not completely recovered along the flight 
line. This is expected since the linear feature is almost 
parallel to the flight direction, which makes it 
ineffective for recovering this component. It only solves 
for the roll angle, while the shift, the scale, the 
direction, and the pitch are solely recovered by the GPS 
observations of the perspective centers. 
  
—— 1.0 m 
^ 
  
  
  
Fig. 1: Error vectors in X and Y using a GPS accuracy 
of 1.0m on the ground and 1.0m at the perspective 
centers. (the errors are in meters, ground units, at all 
points) 
Figure 2 shows the results obtained by 
assuming GPS accuracies of 1.7 m and 1.4 m on the 
ground and at the exposure stations respectively. Figure 
3 presents the errors at the computed tie points under 
the same conditions but with unknown interior 
orientation parameters. It is clear from this figure that 
the system is not capable of solving for the x-coordinate 
of the principal point due to its high correlation with the 
X-component of the exterior orientation parameters. 
Figure 4 shows the results obtained by assuming a GPS 
accuracy of 1.7 m on the ground and 2.0 m at the 
perspective centers. This figure indicates an ill- 
conditioning in the normal equation matrix (i.e. the 
results are not reliable any more). 
Table 1 shows the rms values in X, Y, and Z 
direction for the experiments described by Figs 1 to 4: 
  
  
GPS Accuracy (m) RMS (m) 
Air Ground X Y Z 
1.0 1.0 0.37 0:28 0.45 
1.4 1.7 1.07 1.24 1.86 
1.4 1.7 (sc) 3.95 0.45 6.98 
2.0 1.7 13.42 9.94 14.49 
(sc) self calibration of the camera. 
Table (1), Rms values in X, Y, and Z directions for 
different GPS accuracies both at the perspective centers 
and along the linear feature on the ground. 
207