Page 243 - Innovations in Intelligent Machines

236 J. Gaspar et al.
yields a new shaping function (as in Eq. (13)), which after integrating Eq. (10)
results in the mirror profile proposed by Hicks and Bajcsy [46].
Constant Angular Resolution - One last case of practical interest is that of
obtaining a linear mapping from 3D points spaced by equal angles to equally
distant image pixels, i.e. designing a constant angular resolution mirror.
Figure 4 shows how the spherical surface with radius C surrounding the sen-
sor is projected onto the image plane. In this case the desired linear property
relates angles with image points. Then, placing the constraints into Eq. (9) we
finally obtain Eq. (14).
Integrating Eq. (10), using the shaping function just obtained (Eq. (14)),
would result in a mirror shape such as the one of Chahl and Srinivasan [12].
The difference is that in our case we are imposing the linear relationship
from 3D vertical angles, ϕ directly to image points, (t/F, 1) instead of angles
relative to the camera axis, atan(t/F).
Shaping functions for Log-polar Sensors - Log-polar cameras are imaging
devices that have a spatial resolution inspired by the human-retina. Unlike
standard cameras, the resolution is not constant on the sensing area. More pre-
cisely, the density of the pixels is higher in the centre and decays logarithmic-
ally towards the image periphery. The organisation of the pixels also differs
from the standard cameras, as a log-polar camera consists of a set of concen-
tric circular rings, each one with a constant number of pixels. Advantageously,
combining a log-polar camera with a convex mirror results in an omnidirec-
tional imaging device where the panoramic views are extracted directly due
to the polar arrangement of the sensor.
In a log-polar camera, the relationship of the linear distance, ρ, measured
on the sensor’s surface and the corresponding pixel coordinate, p, is specified
by p = log (ρ/ρ 0 ), where ρ 0 and k stand for the fovea radius and the rate of
k
increase of pixel size towards the periphery.
As previously stated, our goal consists of setting a linear relationship
between world distances (or angles), D and corresponding (pixel) distances,
p. Combining into the linear relationship the perspective projection, ρ = t/F
and the logarithmic law of the log-polar camera, results in the following
constraint:
D = a. log(t/F)+ b (15)

The only difference in the form of the linear constraint when using con-
ventional or log-polar cameras, Eqs. (11) and (15), is that the slope t/F is
replaced by its logarithm. Hence, replacing the slope by its log directly in
Eqs. (12), (13) and (14), results in the desired shaping functions for the log-
polar camera.
Concluding, we obtained a design methodology of constant resolution
omnidirectional cameras, that is based on a shaping function whose speci-
fication allows us to choose a particular linear property. This methodology
generalises a number of published design methods for specific linear proper-
ties. For example the constant vertical resolution design results in a sensor