Intrinsic shape analysis: Geodesic PCA for Riemannian manifolds modulo isometric lie group actions

Abstract

In this paper, we illustrate a new approach for applying classical statistical methods to multivariate non-linear data. In two examples occurring in the statistical study of shape of three dimensional geometrical objects, we illustrate that the current methods of PCA by linear Euclidean approximation are unsuitable if such data in non-linear spaces fall into regions of high curvature, or if they have a large spread. In the following we give an overview of the background of relevant previous work, and an introduction to the building blocks of our work.