Triangulation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 51

Krüger et al. ( 2004 ) for the parabolic peak approximation, its invariance to contrast
and brightness, becomes irrelevant, as both the linear and nonlinear corner models
are about three times more accurate, even for low-contrast images.
Hence, the proposed algorithm allows camera calibration under more complex
conditions and with a larger variety of optical systems than the centre-of-gravity
circular target localisation algorithm or the classical methods for chequerboard cor-
ner localisation.

1.5 Stereo Image Analysis in Standard Geometry

When the intrinsic and extrinsic camera parameters are known or can be inferred
from a reference set of point correspondences, it is advantageous to rectify the stereo
image pair to standard geometry, corresponding to parallel optical axes, identical
principal distances, collinear horizontal image axes, and image planes which are
orthogonal to the optical axes and parallel to the stereo baseline. Accordingly, pairs
of corresponding epipolar lines become parallel to each other and to the horizontal
image axes. The important advantage of performing stereo analysis in image pairs
rectified to standard geometry is the fact that the search for point correspondences
needs to be performed only along corresponding pixel rows of the rectified images.

1.5.1 Image Rectification According to Standard Geometry

In a first step, the original images are warped such that the radial and tangential
lens distortions described by ( 1.3 ) and ( 1.4 ) are compensated. Image rectification
then essentially corresponds to determining a projective transformation for each im-
age such that the conditions stated above are fulfilled. To obtain parallel epipolar
lines, Hartley and Zisserman ( 2003 ) propose a method to project both epipoles to
points at infinity and determine a corresponding projective transformation for each
image based on a set of point correspondences. Their approach does not require the
extrinsic camera parameters to be known. A more compact algorithm for the rec-
tification of stereo image pairs, requiring calibrated cameras, i.e. knowledge about
their intrinsic parameters A i and their extrinsic parameters R i and t i , is introduced
by Fusiello et al. ( 2000 ). Basically, geometric camera calibration is achieved by ap-
plying the principle of bundle adjustment to a reference object of known size and
shape. An overview of geometric camera calibration methods is given in Sect. 1.4 .
In the algorithm by Fusiello et al. ( 2000 ), the image formation process of camera i
is defined by the corresponding projective transformation

P i = A i [ R i | t i ]=[ B i | b i ] with B i = A i R i and b i = A i t i (1.93)

which is shown to yield the relation
b i =− B i W c i (1.94)