Experimental Results and Discussion
In order to determine the accuracy and effectiveness of the
proposed approach, a series of experiments using simulated
and real data has been performed.
Simulated Experiments
Simulated Data
The simulated data is generated by three orthogonal direc-
tions of straight lines and a circle in 3
D
space (Figure 5a).
The 2
D
image (Figure 5b) is projected by using the simulated
camera with 10 mm principal distance. The pixel size is
0.0065(mm/pixels) and the image size is 1280 × 1024 (pixels
× pixels). Gaussian noise with mean 0 and standard devia-
tion 0.2 pixels is added in each image point. The object-space
coordinate system is shown in Figure 5a.
Considering the influence of the first radial distortion coef-
ficient and the distortion center on the results, two series of
experiments with the simulated data have been performed.
As an ellipse is used in the proposed approach, an additional
series of experiments which considers the impact of the circle
size and location has been performed. To compare the results
obtained through the proposed approach, the method based
on line measurements from a single image (van den Heuvel,
1999) has been tested. Moreover, the method without ellipses
has been tested, which shares the same procedures with the
proposed approach except in the second step (i.e., the interior
orientation parameters calculation). In the second step, the
principal distance and principal point are directly calibrated
by the three orthogonal vanishing points. The root mean
square error (
RMSE
) is used to assess accuracies of calibrated
results from the three methods. In addition, the re-projection
errors of all 3
D
points are measured to evaluate the actual
impact of calibration results on 3
D
reconstruction in which
the exterior orientation parameters are calculated from the
vanishing points and the interior orientation parameters
(Gonzalez-Aguilera, 2008).
Experiment 1(Varying the First Radial Distortion Coefficient k
1
)
In the first series of experiments, the first radial distortion
coefficient
k
1
was varied while keeping the principal point
fixed at (
x
0
,
y
0
) = (30, 30) (pixels × pixels). Because the distor-
tion coefficient
k
1
of most modern digital cameras is relatively
weak, the levels of
k
1
were set as ±1.00
E
-07, ±1.00
E
-08, and
±1.00
E
-09 (pixels
-2
). The results of the three methods are
shown in Figure 6a through 6f. In Figure 6, Heuvel represents
the method proposed by van den Heuvel. No_ell represents
the method without ellipses. Pro represents the proposed ap-
proach in this paper.
From the results illustrated in Figure 6, the following con-
siderations can be remarked:
1. As the radial distortion coefficient
k
1
increases, the
errors of the camera parameters
c
,
x
0
,
y
0
,
k
1
obtained
through the method presented by van den Heuvel be-
comes larger. Both the method without ellipses and the
proposed approach can get more stable results than the
method presented by van den Heuvel. This is mainly
due to the use of loop optimization. After calculating
the first radial distortion coefficient and the interior
orientation parameters (the principal distance and the
principal point), the interior orientation parameters
are fed back to the radial coefficient correction. These
(a)
(b)
(c)
(d)
(e)
(f)
Figure 6. Results of experiment 1 using simulated data: (a) through (d) respectively represent the results of the camera parameters
c
,
x
0
,
y
0
,
k
1
with varying
k
1
: and (e) and (f) represent the RMSE of re-projection errors of all the 3D points with varying
k
1
.
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
May 2016
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