CIPA 2003 XIX 1 ' 1 International Symposium, 30 September - 04 October, 2003, Antalya, Turkey
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each image were consecutively introduced into the bundle of six
images as fixed camera parameters.
For all 18 bundle solutions, deviations d x , d Y , d z of all estima
ted tie point coordinates from their known values were calcula
ted, to provide the mean absolute deviations d for all solutions.
This allows assessing the effectiveness of single-image pre-cali
bration using lines. The results are seen in Table 2, in which the
first row gives the outcome of self-calibration.
Table 2. Calibration results from single images
and from bundle adjustment using 6 images
d: mean absolute deviation of tie points from bundle adjustments
using the calibration results from single images as fixed
Image
Method
c (mm)
x 0 (mm)
y 0 (mm)
CT 0
1 d (mm)
Bundle
30.06
-0.18
-0.07
4.4 pm
0.09
A
30.53
-0.54
0.05
6.4 pm
0.09
1
C
30.55
-0.60
0.00
5.7 pm
0.10
B
30.37
-0.12
0.04
3.2
0.09
A
30.67
-0.04
0.36
6.1 pm
0.10
2
C
30.67
-0.04
0.37
6.3 pm
0.10
B
30.81
0.00
0.48
3.2
0.11
A
30.68
0.43
0.61
5.4 pm
0.14
3
c
30.68
0.42
0.62
5.3 pm
0.14
B
30.82
0.46
0.81
3.1
0.15
A
30.79
-0.23
0.19
4.0 pm
0.09
4
c
30.71
-0.19
0.15
3.8 pm
0.09
B
31.91
-0.77
0.39
3.4
0.17
A
30.14
0.20
0.06
6.2 pm
0.11
5
c
29.99
0.28
-0.03
5.2 pm
0.12
B
30.63
0.05
0.26
3.5
0.10
A
30.72
0.80
-0.38
6.0 pm
0.17
6
C
30.74
0.81
-0.38
5.2 pm
0.17
B
29.76
0.26
-0.10
4.9
0.12
Here again, approaches A and C yield very similar results. It is
also seen that all approaches display considerable deviations re
garding the camera constant, whose values consistently exceed
those of the bundle approach. The values for the principal point,
too, show large fluctuations. However, it is basically not advis
able to directly compare parameter values from adjustments dif
fering in input data and/or algorithm. Generally, in bundle solu
tions the camera parameter values are tightly correlated with the
exterior orientation parameters of several images (for instance,
the adjustments of the 6 images and of all 10 images gave a dif
ference of Ac = 0.15 mm for the camera constant).
Thus, object reconstruction with the different camera parameter
values would probably be more reliable for assessing the diffe
rent approaches. In Table 2 it is clearly seen that, for the camera
parameter values obtained from individual images with all three
approaches, the mean absolute differences d (representing accu
racy of the intersected object points) are at most about 1.6 times
larger than those from the self-calibrating bundle solution. One
exception exists in each case (image 6 for approaches A, C and
image 4 for B), with d still being less than 2 times larger.
It may be concluded that, in the present case, single-image ca
mera calibration from linear features results in at most doubling
inaccuracy as compared to self-calibration. This, of course, is an
overall assessment, since approach B appears to behave diffe
rently regarding the individual images.
3.3 Including radial iens distortion in the adjustments
In the preceding tests, lens distortion Ar has been ignored. Now,
it has been included among the unknowns, using Eqs. (2) and
(10) for approaches A and C, respectively. Further, Ar has also
been estimated in a bundle solution using all images. Again, the
values for the camera parameters of each image obtained from
both methods have been introduced, successively, as fixed data
on camera geometry into the bundle adjustment.
The results for distortion, obtained with approach A, are seen in
Fig. 3 (distortion curves from approach C are almost identical).
It is clear that here single-image solutions provide a satisfactory
estimation of radial lens distortion.
Figure 3. Calibrated distortion curves from bundle solution
(dark line) and from vanishing point estimation using Eq. (2).
The results for camera calibration and object reconstruction are
shown in Table 3. In comparison to Table 2, the precision of ad
justments (ct 0 values) has improved considerably thanks to the
correction of distortion; the same holds true for the mean abso
lute differences (d) of tie points.
Table 3. Calibration results from single images and bundle
adjustment with estimation of radial distortion
Image
Method
c (mm)
x 0 (mm)
y 0 (mm)
°0
d (mm)
Bundle
30.06
-0.06
-0.07
3.5 pm
0.05
A
30.47
0.71
0.07
4.9 pm
0.09
C
30.60
1.58
0.03
4.9 pm
0.11
7
A
30.62
0.19
0.33
5.3 pm
0.06
Z
C
30.67
0.26
0.45
5.3 pm
0.07
'l
A
30.38
-0.19
0.31
4.4 pm
0.06
C
30.52
-0.27
0.56
4.4 pm
0.07
7
A
30.76
0.32
0.22
3.4 pm
0.07
C
30.79
0.42
0.26
3.4 pm
0.02
o
A
30.11
-0.10
0.04
4.8 pm
0.05
C
30.11
-0.13
0.12
4.8 pm
0.05
A
30.54
-0.60
-0.32
4.9 pm
0.09
C
30.72
-0.87
-0.42
4.9 pm
0.12
Otherwise, similar remarks as before may be made. Differences
in camera constant remain large, and so does the scatter of the
principal point coordinates. Yet, object point errors from single
image approaches hardly exceed the corresponding value from
bundle solution by a factor of about 2.
4. DISCUSSION
Here, mathematical models for single image calibration, relying
on measurements of straight lines in three orthogonal directions,
have been presented and assessed against bundle adjustment. As
it is, generally, rather unwise to directly compare parameter data
from different sources, evaluation has relied on the use of came
ra information derived from lines as fixed values in bundle solu
tions. Results indicate that, at least in this case, such techniques
lead to ‘reasonable’ errors, about twice as large as those of a ri
gorous solution. In this sense, they could be used when bundle
solutions are impossible or impracticable. Lens distortion, too,
