Aspects of conformal geometric algebra with applications in motion capture

The body of this thesis has three main sections. The first introduces Oriented Conformal Geometric Algebra (OCGA), an extension to conventional Conformal Geometric Algebra (CGA), related closely to Stolfi’s Oriented Projective Geometry. In OCGA the sign of previously homogeneous elements is used to represent a concept of directionality, allowing the definition of half spaces of oriented n-planes or spheres with the boundary element as an oriented n – 1 plane or sphere. Oriented intersection tests allow the order in which a ray intersects a sphere to be identified without a specific test. The utility of this framework is shown in the analysis of a sphere interpolation method of A. Lasenby. The second major section is concerned with the development of a numerically stable general exponential library for CGA bivectors. This work builds on that of Wareham et al. in which a suitable pose bivector exponential expansion was identified. The fully general bivector exponential is derived and the piecewise interpolation paths using pose and dilation bivector exponentials is shown to correspond to a conical helix. The third section considers the problem of taking optical motion capture data, which provides approximate marker locations attached to clothing, to identify and parameterize the underlying human skeleton structure and motion over time. The first of two related chapters is concerned with real-time algorithms suitable for use within a real-time visual feedback system, an example of which is presented. The algorithms presented require 3 markers on each limb segment. Following this is a chapter considering algorithms for offline analysis, either to improve on the accuracy of what is possible in real-time, or to provide estimates with less available data. A new class of symmetric sphere fitting algorithm is introduced.EThOS - Electronic Theses Online ServiceGBUnited Kingdo