Page 241 - Innovations in Intelligent Machines

234 J. Gaspar et al.
expression for F 4 . Re-arranging Eq. (7) results in the following second order
polynomial equation:
2

F +2αF − 1 = 0 (8)
where α is a function of the mirror shape, (t, F) and of an arbitrary 3D point,
(r, z):
− (z − F) F +(r − t) t
α = (9)
(z − F) t +(r − t) F
We call α the mirror Shaping Function, since it ultimately determines the
mirror shape by expressing the relationship that should be observed between
3D coordinates, (r, z) and those on the image plane, determined by t/F.In
the next section we will show that the mirror shaping functions allow us to
bring the desired linear projection properties into the design procedure.
Concluding, to obtain the mirror profile first we specify the shaping func-
tion, Eq. (9) and then solve Eq. (8), or simply integrate:

2
F = −α ± α + 1 (10)
where we choose the + in order to have positive slopes for the mirror shape, F.

Setting Constant Resolution Properties

Our goal is to design a mirror profile to match the sensor’s resolution in order
to meet, in terms of desired image properties, the application constraints. As
shown in the previous section, the shaping function defines the mirror profile,
and here we show how to set it accordingly to the design goal.
For constant resolution mirrors, we want some world distances, D,tobe
linearly mapped to (pixel) distances, p, measured in the image sensor, i.e. D =
a 0 .p+b 0 for some values of a 0 and b 0 which mainly determine the visual field.
When considering conventional cameras, pixel distances are obtained by
scaling metric distances in the image plane, ρ. In addition, knowing that those
distances relate to the slope t/F of the ray of light intersecting the image plane
t
as ρ = f. . The linear constraint may be conveniently rewritten in terms of
F
the mirror shape as:
D = a.t/F + b (11)
Notice that the parameters a and b can easily be scaled to account for a
desired focal length, thus justifying the choice f =1.
We now specify which 3D distances, D(t/F), should be mapped linearly
to pixel coordinates, in order to preserve different image invariants (e.g. ratios
of distances or angles in certain directions).

Constant Vertical Resolution - The aim of the first design procedure is to
preserve the relative vertical distances of points located at a fixed distance,

4
Having an explicit formula for F allows to directly use matlab’s ode45 function