Estimating geodesic barycentres using conformal geometric algebra, with application to human movement




Till, Bernie C.

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Statistical analysis of 3-dimensional motions of humans, animals or objects is instrumental to establish how these motions differ, depending on various influences or parameters. When such motions involve no stretching or tearing, they may be described by the elements of a Lie group called the Special Euclidean Group, denoted SE(3). Statistical analysis of trajectories lying in SE(3) is complicated by the basic properties of the group, such as non-commutativity, non-compactness and lack of a bi-invariant metric. This necessitates the generalization of the ideas of “mean” and “variance” to apply in this setting. We describe how to exploit the unique properties of a formalism called Conformal Geometric Algebra to express these generalizations and carry out such statistical analyses efficiently; we introduce a practical method of visualizing trajectories lying in the 6-dimensional group manifold of SE(3); and we show how this methodology can be applied, for example, in testing theoretical claims about the influence of an attended object on a competing action applied to a different object. The two prevailing views of such movements differ as to whether mental action-representations evoked by an object held in working memory should perturb only the early stages of subsequently reaching to grasp another object, or whether the perturbation should persist over the entire movement. Our method yields “difference trajectories” in SE(3), representing the continuous effect of a variable of interest on an action, revealing statistical effects on the forward progress of the hand as well as a corresponding effect on the hand’s rotation.



Conformal Geometric Algebra, Statistical Analysis, Lie Groups, Euclidean Group