4 - Riemannian geometry on shapes and diffeomorphisms: Statistics via actions of the diffeomorphism group

Abstract

Shapes, which can informally be thought of as curves or surfaces marking the outlines of physical items, are examples of objects that are amenable to statistical analysis using Riemannian geometry. The shapes, either described as continuous curves or surfaces or parameterized as a set of landmark points or an image, can be deformed one to another using smooth invertible functions with smooth inverse (diffeomorphisms). The set of diffeomorphisms is an infinite-dimensional manifold and also a group. We can interpret the variability of shapes by the nature of the deformations applied to them, allowing the shapes to be treated as elements of a highly nonlinear Riemannian manifold.