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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