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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
Several solutions can be applied to overcome this problem. 
First, the translation coefficients may be recovered if the shift 
between the origins of the two coordinate systems is known. In 
this case, the six coefficients representing the scale and rotation 
are recovered by the aid of the GCLs and then the translation 
coefficients (b; and hg) will be determined from the shift 
between the origins. This is a special case, which occurs only 
when local image and object coordinate systems are used. 
Second, the six coefficients relating the scale and rotation 
transformation coefficients are recovered as in the first case and 
a single GCP could be used to define the translation 
coefficients. 
2.1.2 Using the 3D affine LBTM: Similar to the collinearity 
equations in photogrammetry, the 3D affine LBTM can be 
applied to various operations such as space resection for 
calculating the model coefficients (image parameters), space 
intersection for determining the location of a 3D point, and 
image rectification. Recovering the model coefficients for 
images by the LBTM leads to one of the followings processes: 
either working on 2D image rectification (in case of single 
image transformation) or 3D geo-positioning determination (in 
case of using stereo pair images). For image rectification, points 
in image space could be transferred directly to an object plane 
by assuming an average elevation of terrain for the whole area 
covered by the image and by using the recovered model 
coefficients and image coordinates of the points. For the same 
purpose, a Digital Elevation Model (DEM) could also be used 
together with the previous information to enhance the results, 
especially when the model is applied to undulated terrain. 
For 3D geo-positioning determination, stereo images should be 
used. More specifically, a group of GCLs should be used to 
recover the stereo image parameters individually and then 3D 
point coordinates could be calculated. Each point in the area 
covered by the stereo images will raise four equations to 
recover the three unknowns (X,Y,Z), which means one 
redundancy is available to check the quality of the results. 
2.2 The 2D LBTM 
Two-dimensional transformations are required in many 
applications of photogrammetry and remote sensing. We often 
need to convert between coordinates in two different plane 
systems having different origins, orientations, and possibly 
scales. More importance is added to the need for a 2D 
transformation when terrain information is either not necessary 
(as in case of flat terrain or low accuracy requirements) or not 
available (as in case of remote, unmapped areas). 
Similar to the way in which point coordinates in the 3D affine 
LBTM are replaced with unit vectors, the 2D affine LBTM can 
be rewritten as follows: 
a SC ACC EC (10) 
E ^ + 11 
AS Cry FC EC (11) 
where (a,, a,) are unit vector components of a line segment in 
the image coordinate system, (Ax , Ay) are planimetric unit 
vector components of the conjugate line segment in the object 
coordinate system, and C,,...,C; are the model coefficients. 
In addition, the 2D conformal LBTM can be presented for 
unique scale transformation as follows: 
853 
0, C4 —C 4. 4C, (12) 
a=CA +64 +C (13) 
4 
where (a, a,) and (Ay, Ay) are unit vector components of the 
line segment in the image and object coordinate system 
respectively, and C,,...,C, are the model coefficients. 
As was demonstrated in the derivation section of the 3D affine 
LBTM, line unit vectors are not a unique representation of 
lines. Therefore, coefficients presenting the scale and rotation 
can be recovered with the use of GCLs (as was explained in 
section 2.1.1), and calculating the translation coefficients will 
require one additional GCP. A minimum of three GCLs is 
sufficient for the determination of the 2D affine LBTM 
coefficients, and a minimum of two GCLs is enough to 
calculate the 2D conformal LBTM coefficients. Comparing the 
3D LBTM to the 2D LBTM, it can be said that the latter can be 
applied to rectify single images instead of stereo images 
without needing any relief related information. As with the 3D 
LBTM, synthetic as well as real data have been used to verify 
the 2D LBTM model and the results are presented in the 
following section. 
3. EXPERIMENTAL RESULTS AND ANALYSIS 
3.1 Synthetic Data 
The effects of various factors such as differences in terrain 
elevation, inclination angle, length, number, and distribution of 
GCLs were tested for the developed model. Image coordinates 
of four sets (S1, S2, S3, S4) of synthetic stereo image data were 
derived from the actual orientation parameters of a stereo pair 
of Ikonos imagery and synthetic object coordinates of 48 well- 
distributed points. The sets were derived so that both S1 and S3 
represented undulating terrain with height variations of about 
100 m, while S2 and S4 represented flat terrain with height 
variations of less than 15 meters. 
Two different configurations of 12 GCLs were established from 
half of the object points and the remaining object points were 
used as checkpoints. One configuration consisted of long, 200 — 
500 meter, lines with random orientations and the other of 
short, 100 — 200 meter, lines predominately aligned to the 
diagonals of the image. S1 and S2 used the long lines, while S3 
and S4 used the short lines. The two configurations of GCLs 
and checkpoint data are shown in Figures 2 and 3 respectively. 
Extensive sets of experiments were performed, but only a few 
representative cases are reported here due to the space 
limitations. Tables 1 and 2 present the results of the 3D and 2D 
affine LBTM respectively. Different groups of GCLs (from 4 to 
12 GCLs) plus one additional GCP were used, and the results 
are summarized in terms of RMS errors of the 24 independent 
checkpoints in the X and Y directions. The RMS errors in the Z 
direction (in case of using the 3D affine LBTM) are not 
considered here as this study focuses on the rectification of the 
satellite imagery. A detailed discussion of the 3D geo- 
positioning determination will be presented in a future 
publication. 
From the results obtained, it is obvious that the LBTM works 
significantly well for image rectification. The investigation 
shows that, except for the translation coefficients which are 
calculated by the aid of the additional GCP, coefficients