PE&RS May 2016 - page 332

Calibration Toolbox for Matlab is probably the most widely
used open source toolbox for camera calibration nowadays,
which provides an accurate algorithm for camera calibration
based on multiple images of standard chess-board patterns.
Results of Real Images
Table 1 and Table 2 present calibration results from the three
methods for each image, along with the estimated precision of
calibrated parameters, the standard deviation σ 0 of unit weight
a posteriori, and the total number of participating lines and
points. For the proposed approach, the values of σ0 in Table 1
and Table 2 are the standard deviations of the adjustment for
Equation 5. Additionally, the standard deviation of the coef-
ficient k1 is determined in the adjustment of Equation 5. The
ones of the rest parameters are calculated in the adjustments
of the Equation 6. Similarly, Heuvel represents the method
presented by van den Heuvel. Pro represents the proposed ap-
proach in this paper. Bouguet represents the Camera Calibra-
tion Toolbox for Matlab.
To evaluate the actual influence of calibration results on
3D reconstruction, self-calibrating bundle adjustment for 3D
reconstruction has been tested. The 3D points reconstructed
through bundle adjustment, in which the camera parameters
calibrated from Bouguet’s toolbox, were used as the refer-
ence point set. The 3D points reconstruction through bundle
adjustment, in which the calibration results for each test
image were then compared with the reference point set. The
standard deviations σxyz of all the model points reconstructed
for each image are shown in Table 3.
From the results illustrated in Table 1, Table 2, and Table
3, the following considerations can be remarked.
1. Since there are only two vanishing points in the façade
images, the accuracies of calibration results of both the
two methods are relatively lower compared with the
oblique images. Besides, the method presented by van
den Heuvel cannot get principal point coordinates for
the facades image. Still, the proposed approach esti-
mates principal point coordinates by the loop iteration.
Therefore, the proposed approach can obtain more
satisfactory results than the method presented by van
den Heuvel.
2. The accuracies of calibration results for the narrow-
angle images are worse than the ones for the wide-angle
images. Because the longer focal length leads to a degra-
dation of the precision of vanishing points and the im-
ages are taken in the long focal length almost parallel to
the vertical object orientation, the method presented by
van den Heuvel is not suitable for calibration of narrow-
angle images (van den Heuvel, 1999). However, the
accuracies of calibration results through the proposed
approach have been improved to a certain extent.
3. For all the test images, the results of the proposed ap-
proach are much better than the method presented by
van den Heuvel.
Discussion of Experiments
Over the experiments of simulated and real data, it is ob-
served that the proposed method readily identifies the first
radial distortion coefficient and is successful at calibrating the
interior orientation with high accuracy.
For higher levels of the first radial distortion coefficient,
camera parameters are more difficult to calibrate accurately,
but the proposed approach still obtains satisfactory results
for camera calibration. Compared with the method presented
by van den Heuvel, the proposed approach is less affected by
varying radial distortion. Additionally, the method presented
by van den Heuvel does not consider the influence of the
distortion center. On the contrary, the proposed approach
includes the distortion center in the calibration process.
The distortion center is iteratively optimized, which further
refines the interior orientation parameters and the first radial
distortion coefficient. Therefore, the proposed approach is
almost not affected by the different distortion center, while
this factor has a seriously influence on the method presented
by van den Heuvel. The interior orientation parameters are es-
timated through multiple pairs of orthogonal vanishing points
from line segments and an ellipse in the image.
The method presented by van den Heuvel requires a
three-point perspective. However, the proposed approach is
efficient for the estimation of camera parameters in case of
the three-point and two-point perspective. Additionally, the
proposed approach has a better performance of estimating
camera parameters of narrow-angle images. From the results
of all the experiments, there is an obvious improvement over
the method presented by van den Heuvel.
Table 1. Camera Calibration Results for the Wide-Angle I mages
Wide-angle images Method k1 ×10-7 (in pixel-2 ) c (in pixel) x0 (in pixel) y0 (in pixel) σ 0 (in pixel) #lines #points
Oblique
Heuvel -2.02±0.50 799.65±1.03 -8.70±1.95 -20.55±1.04 0.58 328 4796
Pro -2.76±0.39 795.98±0.89 -4.67±0.85 -15.01±0.78 0.45 328 4796
Bouguet -3.07±0.27 793.69±0.62 -5.55±0.59 -11.37±0.77 0.29 - -
Façade
Heuvel -1.76±0.45 738.03±2.31 - - 0.87 140 2140
Pro -2.27±0.30 769.52±1.66 -8.71±0.76 -15.17±1.34 0.61 140 2140
Bouguet -3.07±0.27 793.69±0.62 -5.55±0.59 -11.37±0.77 0.29 - -
T able 2. Camera Calibration Results for the Narrow-Angle Images
Narrow-Angle Images Method k 1 ×10-8 (in pixel-2) c (in pixel) x0 (in pixel) y0 (in pixel) σ 0(in pixel) #lines #points
Oblique
Heuvel -3.02±0.38 1246.89±1.46 11.61±2.54 26.63± 1.22 0.76 210 2,965
Pro -3.72±0.34 1268.68±1.05 4.35±1.59 17.45± 1.14 0.58 210 2,965
Bouguet -3.51±0.36 1286.43±0.44 6.25±0.55 20.72±0.95 0.39 - -
Façade
Heuvel -2.95±0.48 1207.08±2.66 - - 1.29 94 1,310
Pro -3.15±0.40 1237.71±2.15 10.95±2.82 13.17± 1.89 0.83 94 1310
Bouguet -3.51±0.36 1286.43±0.44 6.25±0.55 20.72±0.65 0.39 - -
T able 3. Errors of 3D points Reconstructed for Each I mage
Method
Wide-angle images Narrow-angle images
Oblique
σ xyz (pixels)
Façade σ xyz
(pixels)
Oblique σxyz
(pixels)
Façade σxyz
(pixels)
Heuvel 0.50 1.01 1.13 2.56
Pro 0.42 0.67 0.53 1.54
332 May 2016 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING