Shape-space smoothing splines for planar landmark data

A method is developed for fitting smooth curves through a seriesof shapes of landmarks in two dimensions using unrolling andunwrapping procedures in Riemannian manifolds. An explicit methodof calculation is given which is analogous to that of Jupp &Kent (1987) for spherical data. The resulting splines are calledshape-space smoothing splines. The method resembles that offitting smoothing splines in real spaces in that, if the smoothingparameter is zero, the resulting curve interpolates the datapoints, and if it is infinitely large the curve is a geodesicline. The fitted path to the data is defined such that its unrolledversion at the tangent space of the starting point is a cubicspline fitted to the unwrapped data with respect to that path.Computation of the fitted path consists of an iterative procedurewhich converges quickly, and the resulting path is given ina discretised form in terms of a piecewise geodesic path. Theprocedure is applied to the analysis of some human movementdata, and a test for the appropriateness of a mean geodesiccurve is given.