Triangulation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 2

Chapter 1
Triangulation-Based Approaches
to Three-Dimensional Scene Reconstruction
Triangulation-based approaches to three-dimensional scene reconstruction are pri-
marily based on the concept of bundle adjustment, which allows the determination
of the three-dimensional point coordinates in the world and the camera parameters
based on the minimisation of the reprojection error in the image plane. A framework
based on projective geometry has been developed in the field of computer vision,
where the nonlinear optimisation problem of bundle adjustment can to some ex-
tent be replaced by linear algebra techniques. Both approaches are related to each
other in this chapter. Furthermore, an introduction to the field of camera calibra-
tion is given, and an overview of the variety of existing methods for establishing
point correspondences is provided, including classical and also new feature-based,
correlation-based, dense, and spatio-temporal approaches.
1.1 The Pinhole Model
The reconstruction of the three-dimensional structure of a scene from several images
relies on the laws of geometric optics. In this context, optical lens systems are most
commonly described by the pinhole model. Different models exist, describing op-
tical devices such as fisheye lenses or omnidirectional lenses. This work, however,
is restricted to the pinhole model, since it represents the most common image ac-
quisition devices. In the pinhole model, the camera lens is represented by its optical
centre, corresponding to a point situated between the three-dimensional scene and
the two-dimensional image plane, and the optical axis, which is perpendicular to the
plane defined by the lens and passes through the optical centre (cf. Fig. 1.1 ). The in-
tersection point between the image plane and the optical axis is called the 'principal
point' in the computer vision literature (Hartley and Zisserman, 2003 ). The distance
between the optical centre and the principal point is called the 'principal distance'
and is denoted by b . For real lenses, the principal distance b is always larger than
the focal length f of the lens, and the value of b approaches f if the object distance
Z is much larger than b . This issue will be further examined in Chap. 4 .
In this work we will utilise a notation similar to the one by Craig ( 1989 )for
points, coordinate systems, and transformation matrices. Accordingly, a point x in