Biometrika Advance Access originally published online on August 5, 2007
Biometrika 2007 94(3):513-528; doi:10.1093/biomet/asm047
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Copyright © 2007 Biometrika Trust
Articles |
Shape-space smoothing splines for planar landmark data
Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, Kent CT2 7NF, U.K.
School of Mathematical Sciences, University Park, University of Nottingham, Nottingham NG7 2RD, U.K.
a.kume{at}kent.ac.uk
ian.dryden{at}nottingham.ac.uk
huiling.le{at}nottingham.ac.uk
Received for publication 1 January 2005.
Revision received 1 January 2007.
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Abstract |
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A method is developed for fitting smooth curves through a series of shapes of landmarks in two dimensions using unrolling and unwrapping procedures in Riemannian manifolds. An explicit method of calculation is given which is analogous to that of Jupp & Kent (1987) for spherical data. The resulting splines are called shape-space smoothing splines. The method resembles that of fitting smoothing splines in real spaces in that, if the smoothing parameter is zero, the resulting curve interpolates the data points, and if it is infinitely large the curve is a geodesic line. The fitted path to the data is defined such that its unrolled version at the tangent space of the starting point is a cubic spline fitted to the unwrapped data with respect to that path. Computation of the fitted path consists of an iterative procedure which converges quickly, and the resulting path is given in a discretised form in terms of a piecewise geodesic path. The procedure is applied to the analysis of some human movement data, and a test for the appropriateness of a mean geodesic curve is given.
Key Words: Cubic spline • Shape space • Shape-space spline • Sphere • Spherical smoothing spline • Unrolling • Unwrapping