1197 
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Voi. XXXVII. Part B4. Beijing 2008 
section the method of dynamic epipolar rearrangement was 
introduced. 
The generation of horizontal epipolar images is proposed aimed 
at the situation that the baseline of actual horizontal image is 
tilted [11]. Based on the simple analysis of the basic epipolar 
relations, a general formula of non-horizontal epipolar 
rearrangement in the case of an independent and continuous 
stereopair relative orientation are presented. The problems exist 
in the epipolar rearrangement are divided into two mail aspects: 
(1) Geodetic coordinate system rotation. Under normal 
circumstances, the baseline of the stereopair and geodetic 
coordinate system can not be parallel. If the scan lines of the 
horizontal images are parallel with the epipolar, the geodetic 
coordinate system to the baseline direction in the horizontal 
plane should be rotated. 
(2) Horizontal epipolar rearrangement. Because of the tilt of 
baseline, the epipolar group and the baseline group generated 
from the horizontal images are impossible parallel, so the 
epipolar line rearrangement based on the coplanar conditions 
should still be implemented. 
3.1 Basic analysis relation 
Figure 2. Epipolar line analysis 
As shown in figure 3, for the stereopair with fixed baseline S]S 2 , 
an epipolar plane is constructed with a arbitrary ground point M 
and the baseline. Let the intersecting lines of the epipolar plane 
and the stereopair are / and /’, the project points of M in left and 
right image are P (x p , y p ) and P’. Then SI, S2, P and P’ are 
coplanar, which meets S i S 2 (SPxSP')- Actually, the arbitrary 
point Q in left and right epilolar line (except P) are coplanar 
with the plane constructed with S h S 2 and P. 
Take Si as the origin of coordinate, let the model coordinate of 
S 2 is (B x , B Y , B z ), the image space auxiliary coordinate of Q is 
(X, Y, Z), the image space auxiliary coordinate of P is (X P , Y P , 
Z P ). According to the coplanarity conditions, the points 5/, S 2 , P 
and Q meet the following equation: 
B x B Y B z 
X p Yp Zp 
K X Y Z 
where: 
(XA 
' Xp ' 
Y„ 
= 
Yp 
1-/J 
(x Ì 
Y 
= R 
y 
1-/J 
R 
Q \ a 2 Q 3 
b \ h 2 h 
(2) 
(3) 
(4) 
V c i C 2 c 3y 
where: (x, y) = the image coordinate of Q\ 
f = focal length; 
R l = the rotation matrix of left image; 
R r = the rotation matrix of right image 
R is equal to R L when the Q located in left image. Conversely R 
equal to R r when the Q located in right image, 
3.2 Non-horizontal epipolar images generation 
Based on the coplanar conditions, the generations of non 
horizontal epipolar images get the corresponding epipolar line 
directly in the original images, and then rearrange the unparallel 
epipolar group to parallel epipolar images scan lines. 
For the given baseline component, the rotation matrix of 
stereopair, one point P (x P , y P ) in left image and its 
corresponding image space auxiliary coordinate (X P , Y P , Z P ), 
the epipolar line equation can be derived according to Equation 
1: 
where: 
A C r 
y = —x + — / 
' B B 
(5) 
f = focal length; 
x = image coordinates; 
A - -[m 2 a] - m,b, + (m,k 2 - m^jc,]; 
B = [m 2 a 2 - m,b 2 + (m,k 2 - m 2 ki)c 2 ]; 
C = [m 2 a 3 - m,b 3 + (m,k 2 - m 2 ki)c 3 j; 
m i = B x - B z ki; 
m 2 = By - B z k 2 ; 
k, = X P /Z P ; 
k 2 = Y P /Z P . 
As figure 3 shows, the left end-point of left epipolar lines after 
rearrangement (Figure 3 (b)) have the same coordinates with the 
original epipolar lines (Figure 3 (a)). According to the left end 
points coordinates of left image, the original left and right 
epipolar line equations can be deduced by formula 5. Given 
abscissa x, its vertical coordinate y can be determined. 
By the (x, y), the grey level in the rearrangement epipolar lines 
after performing linear interpolation in y-axis are calculated. 
(a) original image (b) epipolar image 
Figure 3. Relation between original and epipolar images