International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
determined by intersecting the epipolar plane with the 
normalization plane that is horizontal. Alternatively, it can be 
determined by intersecting it with the XY-plane, which results in 
a straight line along the vector (U, V, 0), Figure 7. Therefore, 
the direction of the epipolar lines, x, can be computed as: 
V | N.M'- pe 8) 
K, = arctan|} = 1 = agent —— ( 
U NL+LN 
y 
A 
(LMN) f ee 
   
   
  
MSN?) 
Figure 7. Direction of the epipolar lines 
Utilizing the above findings, the indirect epipolar resampling 
approach of linear array scanner scenes can be summarized in 
the following steps (see Figure 8): 
Left Scene Right Scene 
  
Derive 2-D Affine Parameters & Roll Angle] 
| tL nds an | 
Derive Scene Parallel Parameters: | 
(LM, e» 9. K, Ax, Ay. 5) 
[ PTP Transform 
Derive 2-D Affine Parameters & Roll Angles 
dn dnd ud adadnd ul] 
  
   
Scene el Parameters: | 
(4 M, 0,9. K. dx, dy.) | 
  
  
  
  
'ransformation | 
  
  
| Compute 
0,70,-0 
x, = arctan((N.M'—M.N°)/(N.L'-L.N")) 
Ax, s (Ax4 AYy2S Ay, = (Av + Avy2 
  
  
  
  
  
  
s, =(s+s"}/2 
ee 
À 
| Select Normalized Scene Parameters: | | Select Normalized Scene Parameters: 
D GM rA: a s | Ë Ua MS ) 
| 
ERES, Voi 
[ Re-project the scene ] 
  
Figure 8. Epipolar resampling procedure 
e For each scene, estimate the 2-D Affine parameters, 
together with scanner roll angles using GCP, adopting 
the model in Equations 4. 
e For cach scene, the roll angles are used for PTP 
transformation, and the parallel projection parameters 
are derived from the 2-D Affine parameters. 
e The normalized scene parallel projection parameters 
are obtained by considering a horizontal plane (i.e., 
w,=p,~= 0) and maintaining average scale and shift 
values, and rotating the scenes with an angle x, 
(Equation 8). 
e The scenes are projected from their original parallel 
projection parameters to their normalized values. 
Such a transformation is considered as parallel 
projection between two planes (which is considered a 
6-parameter Affine transformation as discussed in 
Section 3.2). 
5. EXPERIMENTS 
A panchromatic stereopair of IKONOS scenes covering 
Daejeon, South Korea is used in the experiments. The 
geographical coordinates of the area range from 36.26? to 
36.369 North Latitude and from 127.31? to 127.45? East 
Longitude. An overview of these scenes is shown in Figure 9, 
The number of rows and columns and the acquisition data/time 
of the scenes are listed in Table 1. 
    
      
PE 
ene 
  
Left scene Right sc 
Figure 9. Overview of the IKONOS scenes 
  
  
  
  
  
  
  
  
  
Left Right 
Number of rows 13824 14336 
Number of columns 13816 13816 
Acquisition date 2001-11-19 2001-11-19 
Acquisition time 02:18 GMT 02:19 GMT 
  
Table 1. IKONOS scenes’ dimensions and acquisition data/time 
No information regarding the roll angles of the scenes and no 
GCP were available. Instead, the rational function coefficients 
of each of the scenes were provided. They were used to derive 
the ground coordinates of measured tie points - 162 points in 
total (Tao and Hu, 2002). A coordinate transformation was then 
implemented to obtain GCP in the local Cartesian coordinate 
system. Three sets of experiments were tested using different 
numbers of GCP and check points, Table 2. The developed 
approach for epipolar resampling was then performed. The 
square root of the estimated variance components, adopting 
Equations 4, and the average absolute values of the resulting y- 
parallax are listed in Table 2. The means and standard 
deviations of the error values in the object space are also listed 
in the same table. 
  
  
  
  
  
  
  
  
  
  
  
  
  
Experiment 1 2 3 d 
# of GCP 9 25 162 
# of Check points 153 137 0 
G, Left, pixels 4.8 3.7 2.9 
ó, Right pixels 1.7 1.3 1.1 
Mean|P,|, pixels 2.3 1.6 1.5 
GCP Meany+Stdy,,m|0.000+1.707|0.000+0.993 |0.000+0.889 
Mean+Std;, m |0.000+5.674|0.000+6.086|0.000+5.450 
Check|/Meany+Stdyy,m|0.103+1.364|0.095+0.930 - 
points| Mean +Stdz, m | 1.588+6.101|0.587+5.491 - 
  
  
  
  
  
Table 2. Experimental results of the normalization process 
From Table 2, an insignificant improvement between 
Experiments 2 and 3 can be seen. Thus, it can be concluded that 
few GCP can be used for epipolar resampling according to the 
approach developed in this research. In addition, error standard 
deviation values of the check points are not significantly 
different from those of the GCP. Therefore, the suggested 
approaches achieve similar errors throughout the resulting 
International Arch 
HI 
normalized stereo] 
are similar to thos 
are overlaid to g 
which can be stere 
  
Figure 10. St 
6. CONCLU 
In this paper, paral 
scenes such as IK 
model is that ma 
acquired in very 
conform to the 
coordinates along 
transformation so 
The parallel proje 
linear and non-line 
are available while 
are available. A 
combines the linea 
The advantage of t 
estimate the requir: 
the epipolar resamyj 
is briefly presente 
maintains linear ı 
values. Experimen 
feasibility and succ 
Future work will 
scenes. In additi 
epipolar resamplin 
accuracy. Inclusioi 
and areal features) 
projection model 
photos will be gene 
Abdel-Aziz, Y., 
Transformation fr 
Space Coordinai 
Proceedings of 
Photogrammetry, 1 
Urbana, Illinois, pp