Page 235 - Innovations in Intelligent Machines

228 J. Gaspar et al.
mirror profiles, yield simple projection models. In general, to obtain such a
system it is necessary to place the mirror at a precise location relative to
the camera. In 1997, Nayar and Baker [64] patented a system combining a
parabolic mirror and a telecentric lens, which is well described by a simple
model and simultaneously overcomes the requirement of precise assembly.
Furthermore, their system is superior in the acquisition of non-blurred images.
The second design involves specifying a specialised mirror profile in
order to obtain a particular, possibly task-specific, view of the environment.
In both cases, to image the greatest field-of-view the camera’s optical axis is
aligned with that of the mirrors’. A detailed analysis of both the standard
and specialised mirror designs are given in the following Sections.

2.1 A Unifying Theory for Single Centre of Projection Systems
Recently, Geyer and Daniilidis [37, 38] presented a unified projection model
for all omnidirectional cameras with a single centre of projection. They showed
3
that these systems (parabolic, hyperbolic, elliptical and perspective )canbe
modelled by a two-step mapping via the sphere. This mapping of a point in
space to the image plane is graphically illustrated in Fig. 1 (left). The two
steps of the mapping are as follows:
1. Project a 3D world point, P =(x, y, z)toapoint P s on the sphere surface,
such that the projection is normal to the sphere surface.
2. Subsequently, project to a point on the image plane, P i =(u, v)froma
point, O on the vertical axis of the sphere, through the point P s .

Fig. 1. A Unifying Theory for all catadioptric sensors with asinglecentreofpro-
jection (left). Main variables defining the projection model of non-single projection
centre systems based on arbitrary mirror profiles, F(t)(right)

3
A parabolic mirror with an orthographic lens and all of the others with a standard
lens. In the case of a perspective camera, the mirror is virtual and planar.