Algorithm 1 (the steps are illustrated in Figure 3)
1) There exist left and right images. Provided that we want to
generate i-th epipolar line, a point, such as point 1, has to be
chosen. The y coordinate of point 1 in the left image was
assumed in the i-th row (signed y,), while the x, coordinate
can be selected in any position (see Figure 3). For example, if
we want to determine the epipolar line of the first row, y, must
be 1 (row =1), andx, can be selected as any value, e.g. 5
(column —5), or 20 (column 20).
2) Arbitrarily selecting an x coordinate at point 3, (signed
X) (because of coplanarity condition, the equation only
requires that y coordinates be equal, regardless of x
coordinates) in the right image, the y coordinate can be
calculated from Eq. 9.
3) Similarly, select any x coordinates at point 2, (signed x,)
in the left image (selected x, is as far from X, as possible), the
y coordinate of point 2 can be calculated in terms of the
Equation 9 as in step 2.
4) Similar to step 1), calculate the coordinates of point 4
from point 2.
5) Compute the direction of the conjugate epipolar lines by
ki =tan ((y —v,)/64 —x)) (10a)
k, -tan' (03 = Ya) (x5 —X4)) (10b)
where k,,k, denote the slope of the conjugate epipolar lines
in the first and the second images.
6) With the conjugate epipolar lines, we can rearrange the
gray values along the epipolar lines from the original images
by the slopes k,,k,. Nearest pixel interpolation is employed for
this purpose (because it almost does not lose gray information
in resampling process (Lii and Zhang, 1986)).
7) Repeat the steps 1 to 6 until all epipolar lines are
produced.
We summarize the procedures for generation of an epipolar-line
image (including seeking for conjugate epipolar lines,
reassigning gray value along epipolar line) in a stereo pair of
image as follows:
1. Determine more than 8 conjugate points in the left and
right images.
2. Establish the observation equation of Equation 9 using
the conjugate points.
3. Solve the implicit parameters
estimation (LSM).
4. Determine the conjugate epipolar lines, and re-assign the
gray value using Algorithm 1. This step rectifies the original
image into a normal image.
using least sequence
In the above discussion we only take into account a stereo pair.
If each stereo pair is constructed by two neighbor images, and
the geometric rectification of the stereo pairs is carried out
respectively. In this way, all conjugate epipolar lines still
cannot be guaranteed to be in an identical plane, i.e.
coplanarity. This is because these stereo pairs are based on
individual independent rectification coordinate systems. They
have individual original points, and individual u axis. We have
to unify the individual geometric rectifications into the space-
assistance coordinate system. Nevertheless, if we take the first
image as a fixed image, and rectify the other images relative to
it, we can absolutely ensure that all conjugate points/conjugate
epipolar lines lie in an identical plane, i.e. coplanarity. In other
words, we do not need to unify the individual geometric
rectifications into the space-assistance coordinate system.
2.2 Discusses for the geometric rectification algorithm
The constraint, of which all conjugate epipolar lines for any
point, such as p in a sequence of images are coplanar, only
rectifies image distortion along the y direction. Distortion along
the x direction has not been considered above. Additionally, the
geometric distortion caused by deviation from ideal height (in
the vertical plane) cannot yield the y error component in the
image plane. That means this distortion cannot effectively be
rectified in our mathematical model because the y coordinate
component error in the image sequence planes do not vary with
the flying height error. How to rectify two types of distortions
still needs to be studied in future.
B
24 A 6, 5; Ny
16,9) C
Fig.3 Steps illustrated for algorithm 1.
3. EXPERIMENTS AND ACCURACY ANALYSES
3.1 Experiments on geometric rectification
Prof. G. Fisher of the Institute of Space Science at Free
University of Berlin and we built up three test fields in Berlin
(Berlin city, Schónefeld, Werder) to test our algorithm. A
straight-line flight path was flown at a nominal height of 800 m
over the first and the second test fields and at 850m over the
third field. A video camera, S-VHS, Panasonic Video recorder,
was mounted on a CESSNA 207 T platform. Additional details
of the imaging parameters are listed Zhou ef al. (1999, sec
Table 1). We think we closely met the four demands of EPI
analysis.
The original data was recorded on a videocassette, and then
digitally resampled at a frequency of 10 frames/second. The
first frame of each test field is displayed in Figures 4, 5 and 6.
The first test field (see Figure 4) is a simple scene with no
significant high-buildings. The second test field (see Figure 5)
contains an obvious landmark, a chimney, which was used to
test for sensitivity to occlusion and depth discontinuities. The
third test field (see Figure 6) is much more complex, involving
trees, bushes, forest, houses and roads as well as other
buildings.
We developed a system for analysis of the aerial image
sequences, called spatio-temporal technique of aerial image
sequence analysis (STAISA). All of the modules were
programmed in C language in a Silicon Graphics/Indigo Work
Station. Here we only describe the geometric rectification
module, the other modules were described in Zhou et al.,
(1999).
e Feature point extraction
For Equation 9, at least 8 conjugate points need to be extracted
from the image sequences. We used Fóstner operator (Fôstner,
1986) to extract 112 points from the first frame of the first test
field, 214 points from the first frame of the second test field,
and 273 points from the first frame of the third test field. The
black cross in Figure 7 shows the extracted feature points from
the second test field. The magnified window within the Figure
7 illustrates the location accuracy of an extracted feature point.
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