A New Riemannian Setting for Surface Registration

TL;DR

This paper introduces a novel Riemannian framework for surface registration using an inner metric on deformation vector fields, offering a different approach from traditional diffeomorphic matching, with discretization and gradient computation methods.

Contribution

It proposes a new Riemannian surface registration method based on an inner metric on deformation fields, contrasting with existing ambient space deformation approaches.

Findings

The new metric allows direct surface registration in a Riemannian setting.

Discretization of the geodesic equation is achieved via finite elements.

Gradient computation is facilitated for optimization.

Abstract

We present a new approach for matching regular surfaces in a Riemannian setting. We use a Sobolev type metric on deformation vector fields which form the tangent bundle to the space of surfaces. In this article we compare our approach with the diffeomorphic matching framework. In the latter approach a deformation is prescribed on the ambient space, which then drags along an embedded surface. In contrast our metric is defined directly on the deformation vector field and can therefore be called an inner metric. We also show how to discretize the corresponding geodesic equation and compute the gradient of the cost functional using finite elements.