Differential Geometry Tools for Data Analysis

Abstract

The thesis is divided into two parts: the moving anchor parameterization method and the comparative morphology through Willmore-type functionals. In the first part we show that parametrizing points on curves or surfaces or their higher dimensional analogs in Euclidean space by their distances to well-chosen anchor points can lead to representations that are much less curved. We then use this feature to construct approximation methods that are very simple and that achieve results comparable in quality to standard higher-order methods.\\ The second part is motivated by the observation that the Dirichlet normal energy of a 2-dimensional surface embedded in 3D measures, in some sense, the deviation of the surface from the minimal surface fore the Willmore functional. A midified version of this functional with different parameters, is also of interest in applications; this suggests that their minimal surfaces (with respect to which these modified ``Willmore-like''-functionals measure the deviation) are of interest as well. The second part of the thesis concerns the construction of such examples.