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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