Elastic methods for curves in two or more dimensions

We provide statistical analysis methods for samples of curves in two or more dimensions, where only the image but not the parametrisation of the curves is of interest. Examples of such data are handwritten letters, movement paths or outlines of objects. A parametrisation invariant analysis can be based on the elastic distance of the curves modulo warping, but existing methods have limitations in common realistic settings where curves are irregularly and potentially sparsely observed.
We provide methods and algorithms to approximate the elastic distance for potentially sparsely observed curves, useful e.g. for classification or clustering of such curves.
Moreover, we propose to use spline curves for modelling smooth or polygonal Fréchet means of open or closed curves with respect to the elastic distance and show identifiability of the spline model modulo warping.
Finally, we propose a quotient space regression model for elastic regression of such curves on covariates. We test all methods in simulations and apply them to cluster GPS tracks, classify handwritten spirals of Parkinson's patients and controls, and to model how the shape of the human hippocampus is related to age, sex and Alzheimer's disease.