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Title: Smoothing splines on Riemannian manifolds, with applications to 3D shape space Authors:  Ian Dryden - University of Nottingham (United Kingdom) [presenting]
Kwang-Rae Kim - University of Nottingham (United Kingdom)
Huiling Le - University of Nottingham (United Kingdom)
Abstract: There has been increasing interest in statistical analysis of data lying in manifolds. A smoothing spline fitting method is generalized to Riemannian manifold data based on the technique of unrolling and unwrapping originally proposed by Jupp and Kent for spherical data. In particular a fitting procedure is developed for shapes of configurations in general m-dimensional Euclidean space, extending previous work for two dimensional shapes. It is shown that parallel transport along a geodesic on Kendall shape space is linked to the solution of a homogeneous first-order differential equation, some of whose coefficients are implicitly defined functions. This finding enables one to approximate the procedure of unrolling and unwrapping by simultaneously solving such equations numerically, and so to find numerical solutions for smoothing splines fitted to higher dimensional shape data. This fitting method is applied to the analysis of simulated 3D shapes and to some dynamic 3D peptide data.