Shape spaces

Abstract

This chapter describes a selection of models that have been used to build Riemannian spaces of shapes. It starts with a discussion of the finite-dimensional space of point sets (or landmarks) and then provides an introduction to the more challenging issue of building spaces of shapes represented as plane curves. A special attention is devoted to constructions involving quotient spaces, since they are involved in the definition of shape spaces via the action of groups of diffeomorphisms and in the process of identifying shapes that can be related by a Euclidean transformation. The resulting structure is first described via the geometric concept of a Riemannian submersion and then reinterpreted in a Hamiltonian and optimal control framework, via momentum maps. These developments are followed by the description of algorithms and illustrated by numerical experiments.