The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
Pl x l + P2yi + P3 Z 1 + P4 
P9 X 1+ PlOyi + P1I+ 1 
P5 X 1 +P6yi + P7 Z 1 + P8 
P9 X 1+ PlOyi+ Pll +I 
(4) 
where x i, yi, z h x 2 , y 2 and z 2 are the point coordinates in 
space 1 and 2, 
p ,p ,p ,p ,p ,p ,p ,p ,p ,p , andp are the DLT 
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parameters. 
Substituting the values of x 2 and y 2 equation 1 can be written as: 
_ ,Pl x l + P2yi + P3 Z 1 + P4^ 
a 2 ( ) 
P9 X 1 + Ployi + P11 + 1 
, b ^P5 x I + P6TI + P7 z l + P8 
P9 x l + Pl0yi + Pll + 1 
■) + l = 0 
After grouping similar coefficients and normalizing: 
a 2Pl + b 2P5 + P9 
Xj 
<>2P4 +b 2P8 + z l( a 2P3 +b 2P7 + Pll) + 1 
a 2 P2+ b 2P6+PlO 
+ yj 
a2P4 + b 2P8 + z l( a 2 P3 + b 2P7 + Pll) + 1 
+ 1 = 0 
Hence, the relationship between (a^b^and (a 2 , b 2 ) can be obtain 
as shown in equation (5). However, it should be noticed that 
one Z value is used to represent the elevation of the line. This 
implies that only horizontal lines should be used. Or an average 
elevation for the line should be used. 
a 2 Pi + b 2P5 + P9 
a 2P4 +b 2P8 + z l( a 2P3 +b 2P7+Pll) + 1 
a2P2 +b 2P6 + PlO 
a2 P4 + b 2P8 +z l( a 2P3 +b 2P7 +Pll) + 1 
(5) 
where p , p , p , p , p , p , p , p , p , p , and p are the DLT 
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parameters, 
a] and b ( are the line parameters in space 1. 
a 2 and b 2 are the line parameters in space 2. 
independent sets of GCLs, check-lines, and check-points. For 
each line, an average elevation is computed and used to 
represent the elevation of the entire line. 
4.2 Experiments 
Several experiments are conducted to rectify the IKONOS 
image using the transformation models presented in section 3. 
Different combination of GCLs, check-lines, and check-points 
are tested. In the first experiment, four GPS points are used to 
generate 6 GCLs. The remaining eight GPS points are used as 
check-points and are also used to generate check-lines. The 
distribution of GCLs, check-lines, and check-points is shown in 
figure 1. In the second experiment, five GPS points are used to 
generate 10 GCLs. The remaining GPS points are used for the 
checking process. A third experiment was conducted using 15 
GCLs. For each experiment, the transformation parameters of 
each model are computed using the least squares adjustment 
technique. For each pair of corresponding image line and 
ground line, two observation equations are written. The entire 
system of equations is then solved iteratively due to it’s non 
linearity. Approximate values for the transformation parameters 
are initially used to compute the correction to these 
approximate values. The corrected parameters are used to 
compute the line parameters (a and b) of the check-lines. In 
order to facilitate the analysis of the results, the (a and b) 
parameters are converted to the (p and a) parameters. The 
parameter (p) is defined as the length of the perpendicular from 
the origin to the line, while (a) is the angle measured counter 
clockwise from the positive x-axis to that perpendicular. 
Figure 1. GCLs (continuous), check-lines (doted), and check 
points (o) 
4. EXPERIMENT RESULTS AND ANALYSIS 
4.1 Dataset Description 
In this paper, the line-based transformation models presented in 
section 3 are used to rectify the IKONOS image. The proposed 
technique uses only linear features for the rectification process. 
In this research a single Geo panchromatic IKONOS image is 
used. Although the GEO panchromatic image is provided with a 
pixel size of one meter, the absolute positioning accuracy is 
about ±15 meters. GCLs are established using 12 GCPs. The 
points were surveyed using two dual-frequency, Trimble 
4000SS, GPS receivers. The GCPs are used to generate 
4.3 Result Analysis 
Table 1 shows the results of the conducted experiments using 
the presented transformation models. The table shows the 
RMSE of the (p and a) for the check-lines. The results show 
that the RMSE in (p) is less than one meter. In addition, the 
RMSE in (a) is about .05 degree. The X and Y RMSE for the 
check points are less than 1.5 meters for both the 6 and 8 
parameters transformation models. The RMSE for the DLT 
model is larger because the GCLs are not horizontal. 
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