6. TESTS AND ANALYSES
In this section we first describe the implementation
of our algorithm and then report the test results with
an aerial stereo.
The primary step in this algorithm is to determine the
fundamental martix E. As only the ratios among the
entries of E could be determined, we may simply let
one of its component equal to one. It is proper to set
635 = 1 as it is approximately equal to Bx 55 which
may never be zero. All image correspondences are in-
cluded to determine E. Moreover, since there are only
seven degrees of freedom in the fundamental matrix,
the condition
|E| 2 0 (34)
may also be included in the solution procedure
(Barakat et al,1994).
In exterior orientation, the entries of matrix A' is
determined with six known points by Eq.(30). Other
four parameters are then obtained by Eq.(31). After
that the object could be fully reconstructed.
We use an aerial stereo to evaluate our algorithm. Its
primary parameters and the distribution of the six
ground control points (GCPs) are shown in Fig.1 and
Tab.1 respectively
Tab.1 Photographic parameters 1 2 3
Flight height: ca. 2250m
Principle length: 88.94mm
Frame size: 230mm*230mm
Camera: RC-10 3
Overlap: ca. 65% | 4 "B6
Fig.1 GCPs
distribution
Altogether 36 image points as well as their 3D ground
coordinates are measured with an analytical plotter.
The latter are treated as "best values" to check the
validity of our algorithm. Moreover, the DLT algo-
rithm and the traditional collinear algorithm are also
implemented. In order to check the efficiency of our
algorithm, results under different control configura-
tions and various image deformations are presented
respectively in Tab.2 and Tab.3, in which all numerics
are relative to the "best values".
In Tab.2, the DLT algorithm and collinear algorithm
are implemented with all six conjugate GCPs, while
our algorithm is evaluated with different GCPs con-
figuration on the second image and six common GCPs
on the first image.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
It is no wonder that the collinear algorithm holds
the best results (c.f., item Colli. in Tab.2). Item a
and item DLT in Tab.2 show our algorithm obtains
essentially the same rigorous results as the DLT algo-
rithm when they have the same GCPs configuration.
Through item b to fof Tab.2 where the DLT algorithm
is not applicable, our algorithm behaves completely
robust to various GCPs configurations. The small
differences, rarely up to maximum decimeters, are
within the tolerance of GCPs themselves. Moreover,
the most encouraging is in each minimum GCPs con-
figuration we could still reach the same accuracy as
the full GCPs configuration - a benefit due to the
complete employment of the imformation within a
stereo.
Tab.2 Results under different GCPs (in meters)
GCPs config. on | RMSE to best values
the second image | ox oy cz
a: 1-2-3-4-5-6 1.936 | 1.595 | 1:722
b: 2-3-4-6 1.892 11.455 | 1.722
e: 1-2-4-5 1.944 | 1.648 | 1.723
d: 2-3-5-6 1.887 | 1.444 | 1.722
e: 2-4-5-6 1.915 | 1.539 [ 1.722
f: 1-2-3-5 1:054 11.633 | 1.722
DLT algorithm 1.954 | 1.580 | 1.736
Colli. algorithm | 1.376 | 1.365 | 1.745
Tab.3 shows the results of our algorithm under dif-
ferent affine image deformations, where s, a and d
refer to the scale factor, rotation angle and the dis-
parity of the principal point respectively. In order to
testify the validity of our algorithm, simulated affine
deformations based on these parameters are added to
the original image observations, where the first and
second image take different signs of the parameters
respectively. The GCPs configuration for this table
is item b in Tab.2. Since the DLT algorithm presents
the same result under different image deformations, it
is appended there only in the last row.
Tab.3 Results under image deformations (in meters)
Amount of image defor- RMSE to best values
mation parameters ox oy oz
1. no deformation 1.892 | 1.455 | 1.722
2. s=1.1,0 = 10°,d=10mm | 1.926 | 1.460 | 1.715
3. s=0.9,0 = 207,d=20mm | 1.907 | 1.466 | 1.719
4. s=1.3,0 = 30°,d=30mm | 1.923 | 1.465 | 1.748
9.
D
s=0.7,0 = 40°, d=40mm | 1.911 | 1.488 | 1.743
| 1.954 | 1.580 | 1.736 |
LT algorithm
It could be clearly seen that our algorithm is practi-
cally invariant and robust to different amount of affine
image deformations, since only trivial changes (maxi-
mum up to centimeters) might occur among them.
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