Full Stereopsis and Motion from a Coaxial Vertical Pair 

              of Omnidirectional Catadioptric Cameras 

                            Libor Spacek
                        
                       University of Essex
                        spacl@essex.ac.uk  

Abstract:

A convex circular mirror viewed by an ordinary perspective camera (an
omnidirectional catadioptric system) has a number of advantages over
other omnidirectional imaging systems. Its geometry is relatively
simple and the equipment is inexpensive. Unlike rotating
scanner/cameras, the whole image is captured at once and this makes
catadioptric systems more suitable for motion computation, be it of
independently moving objects or an egomotion of an autonomous vehicle.
We use conical mirror because it has less defocus blur in the radial
direction than parabolic, hyperbolic, or elliptical mirrors and
because the conical mirror offers higher resolution around the horizon
where most objects of interest tend to be located. Conical mirrors do
not have single effective viewpoint unless the tip of the cone is cut
off and the center of the camera lens is placed at the (missing)
tip. This means that the epipolar registration is generally regarded
as too difficult for conical mirrors.  However, a particularly simple
solution consisting of radial epipolar lines is obtained when we mount
two catadioptric cameras coaxially (one above the other). In this
special case, the epipolar geometry for the conical mirrors is the
same as for mirrors which do possess the single effective viewpoint.
Therefore the coaxial arrangement of the catadioptric cameras makes
the single effective viewpoint requirement redundant.  We show that by
using a coaxial vertical pair of catadioptric cameras with conical
mirrors we can solve the epipolar stereopsis and hence obtain full 3D
coordinates of surrounding points. In addition, we show that we can
compute full 3D motion vectors of independently moving visual objects,
assuming stationary cameras. We show that, alternatively, we can
compute the 3D translation velocity vector of a moving vehicle,
regardless of its rotation around the vertical camera(s) axis, by
reference to two fixed and well separated visual objects. These
results have many practical applications in 3D vision, robotics and
visual guidance.

Keywords: omnidirectional vision, coaxial catadioptric cameras,
conical mirror, single efective viewpoint, perspective projection,
epipolar geometry, vertical stereo, motion, egomotion.