Conformal Geometric Algebra in Stochastic Optimization Problems ...

232 CHAPTER 9. CONCLUSION
• The method of least squares adjustment and the whole frame of parameter
estimation is illustrated. This includes a survey of the different types of observations
and their corresponding adjustment problems. The focus is on the
linear Gauss-Helmert model, which can account for all types of observations
simultaneously. The estimation method, i.e. the GH-method, that arises from
the model is hence the most general case of least squares adjustment. In the
end, the GH-method for block observations as used throughout this work is
derived.
• The matrix representation and the crucial tensor representation of GA is
explained. On this basis, standard error propagation is adapted to CGA,
that is given a product of two uncertain multivectors the mean and covariance
of the resultant multivector is derived. Error propagation is also applied to
the conformal embedding as it ultimately represents a function of uncertain
arguments.
• Three standard problems are chosen to demonstrate the effectiveness of combining
CGA with the GH-method: first, the estimation of the best circle
passing through a set of uncertain points in 3D. Second, fitting an RBM to
two 3D-point sets, one of which consists of observations. Third, the perspective
pose estimation problem based on point features. Each of these issues
clarifies several important aspects: first, the ease with which such problems
can be modeled if CGA is used. Second, the way the tensor representation
of GA makes algebraic condition and constraint equations available to the
GH-method. Third, how smoothly and with which accuracy error propagation
may be integrated into the framework of geometric algebra. Fourth, the
availability of a covariance matrix for the determined parameters that reflects
how well the estimate approximates the observations.
For each problem, the goodness of the respective GH-solution is experimentally
substantiated.
• Omnidirectional imaging using a single-viewpoint paracatadioptric vision system,
with its strengths and weaknesses, is introduced. A simple method for
calibrating such a system is proposed. Due to its structure, conformal geometric
algebra offers the ideally matching framework to model omnidirectional
imaging in a straightforward manner. This and especially the importance of
the related inversion operation is brought to the fore. To keep track of uncertainties
under omnidirectional image formation, error propagation for CGA
expressions is employed. The GH-method is applied to three problems: a pose
estimation based on point features (a 3D-point model is fitted to projection
rays), a pose estimation based on line features (a 3D-line model is fitted to
projection planes) and an epipole estimation.
For the latter concern, epipolar geometry is entirely modeled within CGA.
As a result, a representation for the essential matrix and the fundamental
matrix, respectively, in terms of CGA elements arises. Epipole estimation
is lastly established on the basis of the essential matrix. This provides, as
a byproduct, the motion estimation between the two considered omnidirectional
images. It is further proven that the renowned authors of [50] make