tios, and the second stereo pair is close to a critical geometry for both algorithms, i.e. the optical axes are almost coplanar the
dihedral angle of their epipolar planes is 1.5° and the distance of the two projection centers from the “ideal” point of intersec-
tion of the optical axes differ only by 2.
Figure 5. Example of simulated data.
4.1 Images with different camera constants
For the estimation of a varying camera constant among the two frames three closed form algorithms were implemented and test-
ed in the experiments:
The algorithm proposed in Section 3.1 which is based on the equality of dihedral angles of epipolar planes
defined independently on both images The algorithm of Newsam et al. 1996, which is ba-
sed on the algebraic properties of the essential matrix The algorithm of Bougnoux 1998, which is based on
the solution of the Kruppa equations An additional non-linear, self-calibrating bundle adjustment so-
lution was also carried out without the use of control points. For initialization the results from the closed form algorithms were
used. In order to check the sensitivity of the proposed algorithms with
respect to errors in the measurement of corresponding image points, normally distributed random errors of various standard
deviations σ
xy
from 0.1 up to 1 pixel were added to the correct image point coordinates. To further check the repeatability of
the algorithms, 20 different solutions were performed for each σ
xy
level. From them a mean c
mean
and a standard deviation c
std
were calculated for the estimated camera constant values. The results of all solutions are presented in Figure 6.
In all experiments, the estimations of the camera constant values from all mentioned closed form algorithms CF in the diagrams
were identical, and at the same time very close to the bundle ad- justment results. This is a confirmation that the algorithm pro-
posed here is equivalent to the ones from the recent Computer Vision literature. The mean values of the camera constant esti-
mations c
mean
are close to ground truth values with differences less than 5. However, it is clear from the standard deviation
diagrams c
std
that the spread of solutions around their mean in- creases with the level σ
xy
of image noise, and so does the uncer- tainty of estimated camera constant values. This is even worse
for the second stereo pair whose image configuration is close to a critical geometry.
750 800
850 900
950 1000
1050
0.1 0.2
0.3 0.4
0.5 0.6
0.7 0.8
0.9 1
σ
xy pixels
c
m e
a n
p ix
e ls
c1 CF c2 CF
c1 Bundle c2 Bundle
20 40
60 80
100 120
140 160
180
0.1 0.2
0.3 0.4
0.5 0.6
0.7 0.8
0.9 1
σ
xy pixels
c
s td
p ix
e ls
c1 CF c2 CF
c1 Bundle c2 Bundle
750 800
850 900
950 1000
1050
0.1 0.2
0.3 0.4
0.5 0.6
0.7 0.8
0.9 1
σ
xy pixels
c
m e
a n
p ix
e ls
c1 CF c2 CF
c1 Bundle c2 Bundle
20 40
60 80
100 120
140 160
180
0.1 0.2
0.3 0.4
0.5 0.6
0.7 0.8
0.9 1
σ
xy pixels
c
s td
p ix
e ls
c1 CF c2 CF
c1 Bundle c2 Bundle
Figure 6. Comparison of different algorithms for the computation of two camera constant values from two simulated
configurations configuration 1 above, configuration 2 below at different noise levels. Mean values and standard deviations
are given from 20 solutions per noise level.
This contribution has been peer-reviewed. The double-blind peer-review was conducted on the basis of the full paper. doi:10.5194isprsannals-III-3-75-2016
80
4.2 Images with common camera constant
