The general setting for Shape Analysis

TL;DR

This paper introduces a unified, rigorous framework for shape spaces in manifold settings, encompassing existing models and enabling advanced analysis methods like LDDMM through sub-Riemannian structures.

Contribution

It provides a general definition for shape spaces in manifolds, unifies various existing models, and formalizes LDDMM methods within a sub-Riemannian geometric framework.

Findings

Shape spaces are formalized as abstract manifolds.

LDDMM methods are characterized as sub-Riemannian structures.

Properties of Hamiltonian geodesic flow are analyzed.

Abstract

In shape analysis, the concept of shape spaces has always been vague, requiring a case-by-case approach for every new type of shape. In this paper, we give a general definition for an abstract space of shapes in a manifold. This notion encompasses every shape space studied so far in the literature, and offers a rigorous framework for several possible generalizations. We then give the appropriate setting for LDDMM methods of shape analysis, which arises naturally as a sub-Riemannian structure on a shape space. This structure is deduced from the space of infinitesimal deformations and their infinitesimal action. We then describe the properties of the Hamiltonian geodesic flow, and study several applications of equivariant mappings between shape spaces.