B3. Istanbul 2004 
Ax 
Ay ( 
0 
nts of interest, P; 
rresponding point, 
object and scene 
>ction vector - i.e., 
and image points, 
rd equation in (1). 
odel is derived by 
‘essed as: 
(2) 
on parameters, and 
no et al., 1999). It 
are useful if the 
other hand, if GCP 
becomes easier to 
ar and non-linear 
ivigation data are 
as to note that a 
mes 6-parameters 
easily seen if we 
ing Z as a linear 
it subsection deals 
» handling scenes 
rmation 
ers conform to the 
n lines. Therefore, 
nodel, Perspective 
ne coordinates are 
scene coordinates 
m to the parallel 
(3) 
line according to 
;pectively. 
a flat terrain. In 
scanner roll angle, 
om the navigation 
roll angle together 
GCP. Combining 
TP transformation 
   
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
xe4Xv-45Y 4.744, 
AX + AY + A,7 + A, (4) 
  
y= 
pend EF Vau E A 
The parameters in Equations 3 - i.e., 4; to Ag and y - can be 
estimated using a minimum of five GCP. This model can be 
considered as a modified version of the parallel projection 
model, by taking into consideration the necessary PTP 
transformation. The next section deals with the utilization of the 
parallel projection parameters for epipolar resampling of linear 
array scanner scenes. 
4. EPIPOLAR RESAMPLING OF LINEAR ARRAY 
SCANNER SCENES 
The epipolar line equation, adopting the parallel projection 
model, can be expressed as (Morgan, 2004; Ono et al., 1999): 
Gx+G,y+G,x+G,y'=1 (5) 
where (x, y) and (x, v) are the coordinates of corresponding 
points in the left and right scenes, respectively. Four parameters, 
G, to G4, express the epipolar line equation. These parameters 
can be derived based on the 2-D Affine parameters or using 
point correspondences. In the latter case, at least four 
corresponding points have to be available. 
From Equation 5, it can be easily seen that epipolar lines are 
straight in the parallel projection model. Furthermore, epipolar 
lines, in any of the scenes, are parallel to each other, Figure 5. 
Therefore, the epipolar line parameters can be utilized to 
transform the original scenes in such a way to eliminate the y- 
parallax values. One can consider rotating the scenes, each with 
different angle, to get the epipolar lines coinciding with the 
scene rows. Afterwards, scale and shift has to be applied to 
obtain corresponding epipolar lines along the same row. Such 
transformation can be expressed as: 
  
I, recu) 
5 2 — 
Jar L À fo 
  
  
  
  
  
Figure 5. Epipolar lines in the parallel projection 
  
i ect = ; 0 
[: |I es (0) sin(0)] x v. A 
JX,] S|-sin(0) cos(9)|| y > (6) 
S: À all . ' ' 0 
es E (0)  sin(0')I x AS 
vd Losin) costo) y] [o 
Where: 
(Xo Fa) are the left scene coordinates after the 
transformation; 
(C, y) are the right scene coordinates after the 
transformation; 
0,0 are the rotation angles of the left and right scenes, 
respectively; 
-4y/2, 4y/2 are the shift values of the left and right scenes, 
respectively; and 
1/8, § are the scale factors of the left and right scenes, 
respectively. 
It is important to note that Equation 5 contains only four 
parameters. Therefore, no more than four parameters can be 
introduced in Equations 6. Eliminating y-parallax (i.e., equating 
y 4 and y,) will result in an equation similar to Equation 5, but 
containing the four parameters 6, 0', Ay, S. By comparing these 
equations, one can derive the following equalities: 
G 
9 =arctan! - — 
5 
  
  
O'= arctan | — S (7) 
4 
s= | Gasin(0) _ [ G,cos(8) 
G, sin (6") G, cos(0") 
AY E sin (0) _ cos (6) _ S.sin (0) | S.cos(9") 
S oco, IPIS z. G, 
In summary, to eliminate y-parallax values, one has to derive 
the epipolar line parameters, G, to G,. Afterwards, the 
parameters (0, 0”, Ay, S) can be computed according to 
Equations 7. Finally, these parameters are used to rotate, scale 
and shift the scene coordinates according to Equations 6. 
Although this procedure is successful in eliminating the y- 
parallax, it is not sufficient to produce useful stereopair. The 
problem is that the resulting x-parallax values are not in linear 
relationship with the height values (Morgan, 2004) By 
analyzing the transformations that were performed (rotation, 
shift and scale), it can be proven that they are planar. In other 
words, these transformations do not change the planes 
associated with the scenes. Recall that the scene planes are 
expressed by the rotation angles (c, 9) in Equation 1. 
To solve this problem, scenes have to be projected onto a 
common plane, similar to what have been discussed in frame 
images in Section 2.1. To select the orientation of this plane, /he 
normalization plane, it is important to remember the main 
objective — that is to have linear relationship between x-parallax 
and height values. Figure 6 shows two points (B and D) of same 
elevation and the parallel projection vectors (L, M, N) and (L”, 
M', N^) of the left and right scenes, respectively. It can be seen 
in this figure that only a horizontal plane (shown as dashed line) 
will maintain similar x-parallax values of points B and D. 
Therefore, scenes have to be projected onto a horizontal plane 
to maintain linear relationship between x-parallax and height. 
N 
d = ; 
N Original left scene 
N 
\h ee Original right scene Zz 
No \ 
\ 
\ Alternative planes Nd 
\p, \d EY c. sacs 
    
(L,M,N) N 
^ 
N EN (LL MNY) 
f \ MN) 
Us Rs \ À 
\ 7 X Sud 
EN T He 
ET 
Figure 6. Selection of the normalization plane 
Following this finding, the direction of epipolar lines within this 
plane can be determined. The parallel projection vectors, 
together with an object point, form a plane in space - epipolar 
plane. The direction of the epipolar lines can then be