Triangulation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 49

Ta b l e 1 . 2 Plot labels and examined methods for chequerboard corner localisation

Plot label Method

Linear Fit a second-order polynomial to low-pass filtered input image (Lucchese and

Mitra, 2002 )

Nonlinear Fit a corner model using Levenberg-Marquardt (this section)

Parabolic Fit a second-order polynomial to correlation coefficient image (Krüger et al., 2004 )

Circle Weighted mean (for circular markers only) (Luhmann, 2006 )

Figs. 1.13 - 1.15 by the vertical extensions of the error boxes. Figure 1.13 a depicts the
overall error over the well-focused images along with the results for a set of blurred
images. The proposed nonlinear fitting algorithm is about two times more accurate
than both the linear algorithm and the centre of gravity-based circle localiser. It is
about three times more accurate than the parabolic approximation of the filter re-
sponse peak. Figure 1.15 b shows that the poor performance of the circle localiser
over all images is caused by its reduced robustness regarding inhomogeneous illu-
mination. Figure 1.15 b also shows that the poor performance of the parabolic peak
approximation is caused in part by its reduced robustness with respect to the offset
from the starting position.
Furthermore, a much better performance of the linear and nonlinear algorithms is
observed when the image is slightly blurred. The performance of the linear and the
nonlinear corner modelling rivals that of the circle localiser for non-blurred high-
contrast images. This is caused by the better signal-to-noise ratio; the location rele-
vant signal (step response) is of lower spatial frequency than in the non-blurred case
and therefore provides more data for the model fit.
Figure 1.13 b shows that the dependence of the performance of the algorithms is
virtually independent of the target position in the image. The slight differences can
be attributed to random variations. Figure 1.14 a shows that the linear and nonlinear
chequerboard corner localisers are basically rotation invariant, as the error values
are nearly identical for all rotations. A slight bias increasing with decreasing angle
can be observed. This trend is of minor importance compared to the error values;
thus it can be attributed to random variations in the data set.
Figure 1.14 b again depicts the superior performance of the nonlinear algorithm
over the linear fit. Even for an angle of only 30 ◦ between the sides of the chequer-
board corner it performs better than the linear fit under any condition. The parabolic
peak approximation again has an error value about three times larger than the other
methods.
Figure 1.15 a shows that the nonlinear fit performs significantly better than both
reference methods under low-contrast conditions. The fact that the centre-of-gravity
circle localiser requires uniform illumination and reflectivity for best results is well
known from the literature (Luhmann, 2006 ). In practise this requires constructive
measures such as retro-reflective targets and confocal illumination. Our experiment
did not provide these in order to investigate the influences of a realistic imaging sit-
uation typically encountered in computer vision scenarios. Although the nonlinear