Sobolev metrics on shape space of surfaces

TL;DR

This paper develops Sobolev inner metrics on shape space of surfaces, ensuring meaningful distances and geodesic equations, with applications demonstrated through numerical solutions.

Contribution

It introduces a general framework for Sobolev inner metrics on shape space, proving non-vanishing distances and well-posed geodesic equations without restrictions on dimensions.

Findings

Sobolev inner metrics induce non-vanishing path-length distances.

Geodesic equations are derived and shown to be well-posed.

Numerical solutions to geodesic equations are successfully computed.

Abstract

Many procedures in science, engineering and medicine produce data in the form of geometric shapes. Mathematically, a shape can be modeled as an un-parameterized immersed sub-manifold, which is the notion of shape used here. Endowing shape space with a Riemannian metric opens up the world of Riemannian differential geometry with geodesics, gradient flows and curvature. Unfortunately, the simplest such metric induces vanishing path-length distance on shape space. This discovery by Michor and Mumford was the starting point to a quest for stronger, meaningful metrics that should be able to distinguish salient features of the shapes. Sobolev metrics are a very promising approach to that. They come in two flavors: Outer metrics which are induced from metrics on the diffeomorphism group of ambient space, and inner metrics which are defined intrinsically to the shape. In this work, Sobolev inner metrics are developed and treated in a very general setting. There are no restrictions on the dimension of the immersed space or of the ambient space, and ambient space is not required to be flat. It is shown that the path-length distance induced by Sobolev inner metrics does not vanish. The geodesic equation and the conserved quantities arising from the symmetries are calculated, and well-posedness of the geodesic equation is proven. Finally examples of numerical solutions to the geodesic equation are presented.