International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
Object space points are simulated and back-projected into 
stereopair using the rigorous perspective projection model. 
Scanner IOP are simulated similar to those of IKONOS (e.g., 
c=10m). EOP are simulated to have almost 100% stereo 
coverage by changing the scanner pitch angles along the flying 
direction, Table 1. It is assumed that the scanner’s trajectory and 
orientation comply with the constant-velocity-constant-attitude 
EOP model. 
Table 4. These values are computed based on the total twenty- 
five points shown in Figure 3. 
PTP transformation is also tested, based on the available 
scanner roll angle. The effect of performing a PTP 
transformation in terms of reduction of the Z-component of the 
errors can be seen. It can then be concluded that omitting such a 
correction results in large errors in the derived height values. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
Table 1. EOP used for the simulation 
Twenty object space points covering an area of 11km by 11km 
are simulated with an average elevation of zero and a height 
variation of +1000m. Figure 3 shows the distribution of the 
object points and the footprint of the left and right scenes. 
Among the object points, sixteen points are used as GCP shown 
as red triangles, while nine points are used as check points 
shown as green circles. 
— Let Scene FootPrnt 
—— Right Scene FoutPrint 
  
4000 £000 4000 2000 2000 4000 6000 8000 
Figure 3. Object space points together with scene footprints 
Using scene EOP in addition to IOP and average elevation, 
scene parallel projection parameters are derived, as explained in 
Section 4 (see Table 2). 
  
  
  
  
  
  
  
Scene X; | Yo | Z | AX | AY | AZ | Roll |Pitch JÁzimuth Meanyy+Stdyy,m Mean +Stdz,m 
Km | Km | Km |Km/s|Km/s|Km/s| v^ | ? 2 No PTP 0.582 + 3.427 0.944 + 0.853 
Left |-288| 60 1680 | 7 0 0 «5 {2251 0 PTP 0.590 + 3.466 0.023 + 0.194 
Right] 277 | -60 | 680 | 7 0 0 5 1225 0 Table 4. Mean error and standard deviation values of the 
directly estimated object space points with and 
without Perspective-To-Parallel (PTP) correction 
For the same synthetic data, the 2-D Affine parameters are 
indirectly estimated using GCP. Three sets of results were 
obtained: without PTP correction; with PTP correction using the 
available roll angle (the true roll angle); and with PTP 
correction using the estimated roll angle (Equations 8). The 
estimated 2-D Affine parameters and the estimated roll angles 
are listed in Table 5. Among the 25 object points, 16 points 
(seen as red triangles in Figure 3) are used as GCP in the 
parameter estimation. The other nine points, shown as green 
circles, are used as check points. 
  
Left scene Right scene 
  
  
PTP- PIP 
No PTP PIE. estimated|No PTP PrP. estimated 
true y true y 
v V 
o: 3.47 2.61 2.60 3.95 2.63 2.58 
pixels 
  
y? 0.00 -5.00 -4.63 0.00 5.00 6.09 
  
A, 1.35e-5 | 1.35e-5 | 1.35e-5 |1.35e-5| 1.35e-5 | 1.35e-5 
  
A, |4.88e-7 | 4.88e-7 | 4.88e-7 |4.88e-7| 4.88e-7 | 4.88e-7 
A, |5.58e-6 | 5.58e-6 | 5.58e-6 _|-5.58e-6/-5.58e-6| -5.58e-6 
A, |3.26e-4|3.26e-4 | 3.26e-4 |-3.26e-4|-3.26e-4| -3.26e-4 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
Table 2. Derived scene parallel projection parameters 
Using the derived scene parallel projection parameters, 2-D 
Affine parameters are derived, as explained in Section 3.3.1 (see 
Table 3). 
  
Scene| A; A» A; 4, [Ast Ae A;  |Ag 
Left | 1.35e-5|4.89e-7|5.59e-6 3.26e-4| 0 | 1.35¢-5|-1.18e-6| 0 
Right| 1.35e-5 |4.89e-7 -5.59e-6|-3.26e-4| 0 |1.35e-5|1.18e-6| 0 
Table 3. Derived 2-D Affine parameters 
  
  
  
  
  
  
  
  
  
  
  
  
  
Some attention must be given to the quality of the derived 2-D 
Affine parameters as well as PTP transformation. Space 
intersection is chosen as a means of examining their quality in 
the object space. In this case, the 2-D Affine parameters for 
both left and right scenes, along with left and right scene 
coordinates, are used. Therefore, for each scene point, two sets 
of 2-D Affine equations are written. For each pair of tie points, 
four equations can be written containing 3 unknowns, the object 
coordinates of the point. Thus, the object coordinates can be 
solved for, in a least-squares adjustment. The errors can then be 
computed in the object space between the estimated coordinates 
and the original coordinates used in the simulation. The mean 
and standard deviation of the errors are computed, as shown in 
Scene, L M aigla|am |dem| 5 As |4.95e-9|4.81e-9 | 4.82e-9 |4.54e-9 | 4.81e-9 | 4.87e-9 
Left |-3.83e-1|8.05e-2| -5 | 0 | 0 |3.26e-4| O |1.35e-5 As |1.35e-5|1.35e-5 | 1.35e-5 | 1.35e-5) 1.35e-5 | 1.3565 
Right| 3.83e-1 |-8.05e-2| 5 0 | O |-3.26e-4| 0 |1.35e-5 A;  -1.20e-6|-1.19e-6| -1.19e-6 | 1.18e-6| 1.17e-6 | 1.17e-6 
  
  
  
  
  
  
  
  
As | 1.62e-5 |-7.80e-6| -6.03e-6 |-3.50e-5|-1.13e-5| -6.06e-6 
Table 5. Indirectly estimated 2-D Affine parameters and roll 
angles using GCP 
  
As seen in Table 5, the square root of the estimated variance 
component, g,, is the smallest by using the estimated roll 
angles for PTP correction. On the other hand, omitting PTP 
correction results in the largest estimated variance component. 
In addition, the estimated roll angles (as indicated in Equation 
8) differ from the true roll angles. The difference is attributed to 
the assumption made in Section 3.4 of a flat terrain. On the 
other hand, it can be seen that using the estimated roll angles 
gives better results, in terms of the smallest variance 
component. In order to evaluate the quality of the estimated 2-D 
Affine parameters, a space intersection is performed based on 
the estimated parameters and the coordinates of the points in 
both left and right scenes, as explained earlier. The estimated 
object coordinates are then compared to the true values and the 
errors associated with the points in Tables 6. 
  
International Ai 
     
No | 
PTP | 
PTP esti 
; No] 
i. PIP 
PTP esti 
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Abdel-Aziz, Y. 
Transformation 
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Proceedings of 
Photogrammetry, 
Urbana, Illinois, | 
Dowman, I., and 
Functions for | 
Archives of Photo 
254-266.