Riemannian geometry for shape analysis and computational anatomy

TL;DR

This paper provides an accessible overview of the use of Riemannian geometry and functional analysis in shape analysis and computational anatomy, focusing on infinite-dimensional manifolds and Sobolev metrics.

Contribution

It offers a comprehensive roadmap explaining how differential geometry and functional analysis interact in the context of shape analysis and computational anatomy.

Findings

Clarifies the mathematical foundations of shape analysis

Connects differential geometry with functional analysis in applications

Provides guidance for researchers new to the field

Abstract

Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations exist, it is sometimes difficult to gain an overview how differential geometry and functional analysis interact in a given problem. This paper aims to provide a roadmap to the unitiated to the world of infinite-dimensional Riemannian manifolds, spaces of mappings and Sobolev metrics: all tools used in computational anatomy and shape analysis.