Conformal Geometric Algebra in Stochastic Optimization Problems ...

Chapter 7
Applications in Computer
Vision
This is the first time that all concepts of this thesis are being brought together.
Three different parameter estimation examples shall demonstrate the goodness of
the previously derived GH-method, see preferably section 5.3.4, when applied to
problems expressed in the language of conformal geometric algebra. It is begun with
a simple example of fitting a circle to a set of uncertain points in 3D, which was
also one of the first experiments with the GH-method. The elucidations culminate
in the presentation of a solution to the perspective pose estimation problem as
introduced in chapter 4. The subsequent descriptions clearly builds on CGA, but
subjects such as error propagation are prerequisites as well; refer to chapter 6 in
this respect. In the end, experimental results including comparisons to standard
approaches are presented.
Recall that, in general, the aim is to find multivectors that satisfy a particular
condition that depends on a set of uncertain measurements. The specific problem
and the type of multivector, representing a geometric entity or a geometric operator,
determine this condition. Here point measurements from Euclidean 3D-space are
considered, where the respective uncertainties are assumed to be given by covariance
matrices.
Related Work
A discussion regarding the linear estimation of rotation operators in geometric algebra
can be found in [98], albeit without taking account of uncertainty. In the scope
of perspective pose estimation Rosenhahn and Sommer [104] derived a method for
estimating rotation/translation operators by means of conformal geometric algebra.
Their approach is mainly based on the stratification hierarchy of Euclidean, projective
and affine spaces. Based on previous works by Förstner et al. [35] and Heuel
[65], where uncertain points, lines and planes were treated in a unified manner, the
estimation of uncertain CGA operators was introduced in [95], which can be viewed
as a foundation for this text.
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