Conformal Geometric Algebra in Stochastic Optimization Problems ...

220 CHAPTER 8. APPLICATIONS IN OMNIDIRECTIONAL VISION
when the sensor approaches an object it becomes taller - the diametrically opposite
object becomes smaller. When the senor moves, the image points move along the
epipolar circles, see figure 8.16.
Each line in the top left image of figure 8.13 is sampled from the lower half of an
epipolar circle between the epipole and its diametrically opposite counterpart. Such
rectifications are usually employed in stereo vision because any customary scan line
based stereo matching algorithm can then compute the corresponding disparities
which ultimately provide the 3D-reconstruction of a scene. Nevertheless, it must
be taken into account that only the green-rimmed area in the figure can effectively
be used.
Fig. 8.13: On the role of the epipoles.
8.4.2 Discovering Catadioptric Stereo Vision with CGA
Now a condition for the matching of image points is formulated. This enables the
derivation of the fundamental matrix F and the essential matrix E for the parabolic
catadioptric case.
Consider the stereo setup of figure 8.14 in which the imaging of world point Pw
is depicted. Each of the projection spheres S and S ′ represents the catadioptric
imaging device, but at different positions. The interrelating RBM indicates the
sensor movement - from the original to the primed coordinate system. The centers
of the coordinate systems are assumed to coincide with the respective focal points,
i.e. F and F ′ . Note that the (left) primed coordinate system is also rotated about
the vertical axis.
The two projections of Pw are X and Y ′ . Let x and y be their corresponding
image points, given as conformal embeddings {x,y} = K({�x,�y} ⊂ � 3 ) of the pixel
coordinates w.r.t. the image center F. The inverse stereographic projection (from
the plane to the sphere) of x and y yields the points X and Y , represented in the
unprimed coordinate system. In order to do stereo, considerations must involve the
RBM, which is denoted by M. Hence one can write
Y ′ = MY � M,