Page 237 - Innovations in Intelligent Machines

230 J. Gaspar et al.
2
centre system combining a parabolic mirror, F(t)= t /2h with an ortho-
graphic camera [65], one obtains the projection equation, φ =2atan(t/h)
relating the (angle to the) 3D point, φ and an image point, t.
In order to make the relation between world and image points explicit it is
only necessary to replace the angular variables by cartesian coordinates. We
do this assuming the pin-hole camera model and calculating the slope of the
light ray starting at a generic 3D point (r, z) and hitting the mirror:

! " ! "
t r − t
θ = atan , φ = atan − . (3)
F z − F
The solution of the system of equations (2) and (3) gives the reflection point,
(t, F) and the image point (f.t/F, f) where f is the focal length of the lens.

2.3 Design of Standard Mirror Profiles

Omnidirectional camera mirrors can have standard or specialised profiles,
F(t). In standard profiles the form of F(t) is known, we need only to find
its parameters. In the specialised profiles the form of F(t)isalsoadegreeof
freedom to be derived numerically. Before detailing the design methodology,
we introduce some useful properties.
Property 1 (Maximum vertical view angle) Consider a catadioptric
camera with a pin-hole at (0, 0) and a mirror profile F(t), which is a strictly
positive C 1 function, with domain [0,t M ] that has a monotonically increasing
derivative. If the slope of the light ray from the mirror to the camera, t/F is
monotonically increasing then the maximum vertical view angle, φ is obtained
at the mirror rim, t = t M .
Proof: from Eq. (2) we see that the maximum vertical view angle, φ is
obtained when t/F and F are maximums. Since both of these values are

monotonically increasing, then the maximum of φ is obtained at the maximal
t, i.e. t = t M .

The maximum vertical view angle allows us to precisely set the system
scaling property. Let us define the scaling of the mirror profile (and distance
.
to camera) F(t)by(t 2 ,F 2 ) = α.(t, F), where t denotes the mirror radial coor-
dinate. More precisely, we are defining a new mirror shape F 2 function of a
new mirror radius coordinate t 2 as:
. .
t 2 = αt ∧ F 2 (t 2 ) = αF(t). (4)

This scaling preserves the geometrical property:
Property 2 (Scaling) Given a catadioptric camera with a pin-hole at (0, 0)
andamirrorprofile F(t),which is a C 1 function, the vertical view angle is
invariant to the system scaling defined by Eq. (4).