Conformal Geometric Algebra in Stochastic Optimization Problems ...

222 CHAPTER 8. APPLICATIONS IN OMNIDIRECTIONAL VISION
6.1 can be exploited such that every pair of correspondence points (x,y) yields the
G-condition
g t (p, x,y) = x k O t kc (p r G a rl y l G c ab R b s p s ), (8.8)
where Φ(G) = g, Φ(X) = x, Φ(Y ) = y and Φ(M) = p. As usual, the product
tensors O, G and R denote the outer product, the geometric product and the reverse,
respectively. Likewise, for the motor M the familiar parameterization p ∈ � 8 is
chosen.
Note that only a particular index t = t ⋄ , that is the one indexing to the e ∗ ocomponent
of the result, has to be taken into account. After setting F = eo it
can be shown that x and y in fact denote the Euclidean 3D-coordinates on the
projection spheres, i.e. x, y ∈ � 3 . Considering the motor p as constant, the bilinear
form
g(x,y) = x k Ekl y l ∈ �
with
Ekl = O t⋄
kc p r G a rl G c ab R b s p s
is obtained. The condition is linear in X and linear in Y as expected by the
bilinearity of the geometric product. Its succinct matrix notation is
x T Ey = 0 , (8.9)
where E ∈ � 3×3 denotes the essential matrix of the epipolar geometry. No proof
is given, but it is mentioned that equation (8.9), which ultimately reflects a triple
product, is zero if and only if there is coplanarity between the four points F ′ , Y ′ ,X
and F. Next, if setting Y ′ = E ′ or X = E one gets E y = 0 and x T E = 0,
respectively. Otherwise, say Ey = n ∈ � 3 , an X can be chosen such that the
corresponding x is not orthogonal to n, whence x T Ey �= 0. This would imply
that the points F ′ , E ′ , F and the chosen X are not coplanar, which must be a
contradiction since F ′ , E ′ and F are already collinear. Hence the 3D-epipoles reflect
the left and right null space of E, and it can be inferred that the rank of E can be
at most two.
Because E does solely depend on the motor M, which embodies the extrinsic parameters,
it can not be a fundamental matrix, which must include the intrinsic
parameters as well. Fortunately, the previous derivations can easily be extended
to obtain the fundamental matrix F. Recall the image points x and y. They are
related to X and Y in terms of a stereographic projection. As already stated in
section 8.1, a stereographic projection is equal to an inversion in a certain sphere,
but inversion is the most fundamental operation in CGA. In accordance with figure
8.2 it can be used X = SIxSI. Note that the inversion sphere SI depends on the
focal length of the parabolic mirror. In this way equation (8.7) becomes
G = F ∧ (SIxSI) ∧ F ′ ∧ (SIy ′ SI) e∗o = 0.
In addition, the image center (specifically, the coordinates of the pixel where the optical
axis hits the image plane) can be included by introducing a suitable translator
TC. Hence SI would have to be replaced by the compound operator Z := SITC
G = F ∧ (Zx � Z) ∧ F ′ ∧ (Zy ′ � Z)
e ∗ o
= 0. (8.10)