Page 236 - Innovations in Intelligent Machines

Toward Robot Perception through Omnidirectional Vision 229
The mapping is mathematically defined by:

u l + m x
2
= , where r = x + y + z 2 (1)
2
v l · r − z y
As one can clearly see, this is a two-parameter, (l and m) representation,
where l represents the distance from the sphere centre, C to the projection
centre, O and m the distance from O to the image plane. Modelling the various
catadioptric sensors with a single centre of projection is then just a matter
of varying the values of l and m in 1. As an example, to model a parabolic
mirror, we set l =1 and m = 0. Then the image plane passes through the
sphere centre, C and O is located at the north pole of the sphere. In this
case, the second projection is the well known stereographic projection. We
note here that a standard perspective is obtained when l =0 and m =1. In
this case, O converges to C and the image plane is located at the south pole
of the sphere.

2.2 Model for Non-Single Projection Centre Systems

Non-single projection centre systems cannot be represented exactly by the
unified projection model. One such case is an omnidirectional camera based
on an spherical mirror. The intersections of the projection rays incident to the
mirror surface, define a continuous set of points distributed in a volume[2],
unlike the unified projection model where they all converge to a single point.
In the following, we derive a projection model for non-single projection centre
systems.
The image formation process is determined by the trajectory of rays that
start from a 3D point, reflect on the mirror surface and finally intersect with
the image plane. Considering first order optics [44], the process is simplified to
the trajectory of the principal ray. When there is a single projection centre it
immediately defines the direction of the principal ray starting at the 3D point.
If there is no single projection centre, then we must first find the reflection
point at the mirror surface.
In order to find the reflection point, a system of non-linear equations can
be derived which directly gives the reflection and projection points. Based on
first order optics [44], and in particular on the reflection law, the following
equation is obtained:
φ = θ +2.atan(F ) (2)

where θ is the camera’s vertical view angle, φ is the system’s vertical view
angle, F denotes the mirror shape (it is a function of the radial coordinate,
t)and F represents the slope of the mirror shape. See Fig. 1 (right).

Equation (2) is valid both for single [37, 1, 96, 82], and non-single pro-
jection centre systems [12, 46, 15, 35]. When the mirror shape is known, it
provides the projection function. For example, consider the single projection