Triangulation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 41

formations and nonlinear distortions. By contrast, circular dots are influenced by
these effects. A simple explanation for this behaviour relies on the fact that chequer-
board corners are scale invariant as long as the localisation window only contains
the corner. The coordinate of interest is the intersection of the edges of the four
adjacent chequerboard fields, thus a point. The projection of a point is invariant to
the aforementioned influences, as it remains a point. In contrast, the centre of a pro-
jected circle is not necessarily found at the same image position as the projection of
the centre of the observed circle; thus circular targets suffer from a bias in their cen-
tre coordinates. This effect becomes increasingly pronounced when lens distortion
effects become stronger.
Mallon and Whelan ( 2006 ) present an edge-based nonlinear chequerboard cor-
ner localiser. Their method performs a least-squares fit of a parametric model of the
edge image with the edge image itself. The described approach makes use of the
method by Li and Lavest ( 1995 ), where it is applied to calibration rigs consisting
of white lines on a black background (KTH calibration cube). The drawback of this
method is its dependence on edge images; it requires the computation of the edge
image of the input image, which is an additional (but small) computational bur-
den. Furthermore, the original method by Li and Lavest ( 1995 ) uses a rather ad hoc
approach to the modelling of the corner image, as it relies on 11 parameters com-
pared to our 7 parameters. The 4 additional parameters account for an illumination
gradient, different reflectivities of the white lines, and different line widths. These
additional parameters put an unnecessary computational burden on the optimiser.
In general more model parameters increase the probability of getting stuck in local
minima.
One of the popular algorithms for finding the centre of a circular white target
is the computation of the centre of gravity of a window around the detected posi-
tion (Luhmann, 2006 ). Here, the square of the grey value serves as the weight of
a pixel. The ensuing algorithm is simple and fast. It is, however, sensitive to inho-
mogeneous grey values caused e.g. by defects of the target or non-uniform incident
light. We will use this method as a reference in this study.
Locating a chequerboard corner can be achieved by various methods. Chen and
Zhang ( 2005 ) use the intermediate values computed in the corner detection phase to
obtain the second-order Taylor expansion of the image and find its saddle point. The
underlying corner model is restricted to orthogonal corners. The result is invariant to
rotation, translation, contrast, and brightness, but not to projective transformations
or nonlinear distortions.
The method by Lucchese and Mitra ( 2002 ) performs a least-squares fit of a
second-order polynomial to a low-pass version of the input image. The position of
the saddle point is obtained from the polynomial coefficients. The underlying model
is invariant to affine transformations, contrast, and brightness. If the neighbourhood
of the corner is chosen small enough, the affine invariance yields a suitable approx-
imation in the presence of projective transformations and nonlinear distortions. We
will use this method as a reference in this paper. The method is able to cope with the
distortions introduced by fisheye lenses, but requires a careful tuning of the size of
the low-pass filter and at the same time the window size for fitting. The low-pass pa-
rameters depend mostly on the width of the point spread function (PSF) of the lens.