Conformal Geometric Algebra in Stochastic Optimization Problems ...

Chapter 9
Conclusion
It was intended to demonstrate that combining conformal geometric algebra with
the method of least squares adjustment is best suited for a wide range of applications,
especially those from the field of computer vision. The main topic dealt
with from this subject is pose estimation. But the individual contributions of the
chapters shall be subsumed first.
• A thorough and formal introduction to GA with vivid examples and several
connections to the standard vector algebra is given. The outer product and its
momentous consequence - the blade - is derived in every detail. Sophisticated
algebra expressions are analyzed and broken down into intelligible representations.
Operations as the reverse, the magnitude, the conjugate, the inverse,
the projection or the rejection are explained. Vital concepts as duality and
outermorphism are elucidated. The relevance of versors and null blades is
discussed.
• The conformal space and its underlying embedding is illustratively derived.
Conformal geometric algebra, its rich subspace concept and the transformations
from the conformal group, which can act on the geometric objects that
live in the subspaces, are enlightened. The multivectors inhering with these
properties are analyzed in detail with respect to their structure and their mutual
relationships. An entirely algebraic factorization of motors (rigid body
motions) into specific translational and rotational parts is proposed.
• The principle of pose estimation is succinctly rephrased using CGA. Starting
from a clear geometric concept, a new geometric view on the 3-point pose
estimation is obtained. The solution so inspired amounts to finding a root of a
well-behaved scalar valued function of an angle. An n-point problem is shown
to be solvable by taking the algebraic group nature of motors into account;
a technique called intrinsic mean builds a weighted average of several 3-point
solutions exploiting the tight relationship to the Lie algebra of the motors.
The resultant method is robust, sound and provides accurate estimates.
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