Triangulation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 42

Krüger et al. ( 2004 ) utilise intermediate information of the corner detector. A cor-
relation coefficient between the image and a corner template is computed for each
pixel. Peaks in this correlation coefficient image indicate the presence of chequer-
board corners. The subpixel accurate position is obtained by computing the position
of the local extremum by fitting a second-order polynomial. Incidentally, this op-
eration is identical to the method by Lucchese and Mitra ( 2002 ), but it operates on
different input data. For this reason we include the method by Krüger et al. ( 2004 )
as another reference approach.
In the following section we derive a chequerboard corner model and fitting pro-
cedure that copes with strong distortions and different PSF widths while providing a
high accuracy. As a by-product we obtain the radius of the PSF along with the target
position.
Olague and Hernández ( 2005 ) present a technique that can be seen as a precursor
to the method proposed in this study. They model an L-corner by two overlapping
smoothed step functions. They use a Gaussian kernel as the smoothing operator and
therefore model their step function by the Gaussian error function. We are using
a similar step function model in our approach, but provide an additional approxi-
mation. Since the Gaussian error function is provided by most numerical libraries
as a piecewise polynomial approximation, it is quite slow to compute. We propose
using the sigmoid function, which is based on a single exponential that exists as
a floating-point operation in many processor architectures and is therefore fast to
compute. Furthermore, we use chequerboard corners instead of L-corners and as-
sume a circular PSF, thus reducing the number of parameters to 7, compared to 12
by Olague and Hernández ( 2005 ).
We will compare our approach to the methods by Lucchese and Mitra ( 2002 ),
Krüger et al. ( 2004 ), and the centre of gravity method according to Luhmann ( 2006 )
for circular targets. The comparison between accuracies achieved with the same
camera system under identical illumination conditions for corner targets and cir-
cular targets is of high interest due to the fact that circular targets are commonly
used for camera calibration in photogrammetry. In the literature, such comparisons
are generally performed using synthetically generated images (Luhmann, 2006 ). In-
stead, we rely on a set of real images labelled with the displacement in metric units,
where the independently determined pixel scale is used to compute the displacement
in pixels as a ground truth. Our direct comparative method differs from previous in-
direct evaluation approaches that compare the reprojection error of a subsequent
bundle adjustment stage (Triggs et al., 2000 ; Salvi et al., 2002 ; Luhmann, 2006 ),
the three-dimensional reconstruction error of selected points in the scene (Salvi et
al., 2002 ), or the variances of camera parameters obtained by a subsequent extrinsic
calibration stage (Olague and Hernández, 2005 ).

1.4.8.2 A Model-Based Method for Chequerboard Corner Localisation

We regard a square image window I ∗

1 ) pixels.
The central pixel is assumed to have the coordinates ( 0 , 0 ) and to be the location

with a width and height of ( 2 r +