Triangulation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 50

fit results in a higher error value and a slight bias due to the inhomogeneous illu-
mination, the performance drop is not as pronounced as for the reference methods.
The nearly quadrupled error of the circle localiser along with a bias larger than the
error of the corresponding high-contrast result illustrate the superior robustness of
the nonlinear fit, which shows only an insignificant bias.
The peak approximation is not significantly influenced by the contrast, as the
correlation coefficient is brightness and contrast invariant. Even in the low-contrast
setting the peak approximation is outperformed by both the linear and the nonlinear
corner models.
Figure 1.15 b shows that three of the examined algorithms are robust to variations
of the initial position in a 3

3 neighbourhood. The peak approximation shows a
clear trend regarding its error and the accuracy of the initial corner detector. If the
corner detector is off by only 1 pixel diagonally, the error increases by about one
third.
At this point, note that all stated accuracies refer to displacements, and thus the
differences of target positions. According to the law of error propagation (Bronstein
and Semendjajew, 1989 ) and assuming identical error distributions for the position
measurements used to determine a set o f differences, the actual errors of the corner

×

positions are smaller by a factor of 1 / √ 2 than the displacement errors stated above.

1.4.8.4 Discussion

The nonlinear chequerboard corner localisation algorithm proposed by Krüger and
Wöhler ( 2011 ) attains an overall error value (half the difference between the 75 %
and 25 % quantiles) of 0 . 032 pixel averaged over image contrast, target rotation,
shear, and position in the image. The error values are 0 . 024 and 0 . 043 pixel for high-
and low-contrast images, respectively, and becomes as low as 0 . 016 pixel for slightly
blurred images. For comparison, the classical photogrammetric method based on
circular targets achieves 0 . 045 pixel in the average case, and 0 . 017 pixel for high
and 0 . 132 pixel for low image contrast. All these error values refer to differences
between pairs of corner positions. No evidence was found for systematic errors for
any of the examined chequerboard corner localisation methods.
The proposed algorithm is also suitable to evaluate the focusing of lenses and
would allow the calibration of the Depth-Defocus-Function (Kuhl et al., 2006 )for
the computation of depth from defocus based on the estimation of the parameter

σ

in ( 1.89 ).
The evaluation based on real test images has shown that the proposed nonlinear
algorithm is more robust with respect to unfavourable imaging conditions (e.g. low
image contrast) than the linear algorithm by Lucchese and Mitra ( 2002 ) and the
centre-of-gravity-based localiser for circular targets according to Luhmann ( 2006 ).
Furthermore, it can be applied earlier in the image processing chain than a model-
fitting circle localiser, e.g. as proposed by Heikkilä and Silvén ( 1997 ), as it is inde-
pendent of the camera model. A further advantage over the linear algorithm is that
the input images do not need to be low-pass filtered. The main advantage stated by