Multivariate Hermite interpolation of manifold-valued data

TL;DR

This paper introduces two novel methods for multivariate Hermite interpolation of manifold-valued data, one using weighted Riemannian barycenters and the other employing tangent space techniques, both validated through numerical experiments.

Contribution

It presents two new intrinsic methods for manifold-valued Hermite interpolation, avoiding local coordinates and embeddings, and compares their effectiveness.

Findings

Both methods effectively perform Hermite interpolation on manifolds.

The Riemannian barycenter approach requires solving linear systems without vector transport.

The tangent space approach depends on the choice of base point and uses vector transport.

Abstract

In this paper, we propose two methods for multivariate Hermite interpolation of manifold-valued functions. On the one hand, we approach the problem via computing suitable weighted Riemannian barycenters. To satisfy the conditions for Hermite interpolation, the sampled derivative information is converted into a condition on the derivatives of the associated weight functions. It turns out that this requires the solution of linear systems of equations, but no vector transport is necessary. This approach treats all given sample data points equally and is intrinsic in the sense that it does not depend on local coordinates or embeddings. As an alternative, we consider Hermite interpolation in a tangent space. This is a straightforward approach, where one designated point, for example one of the sample points or (one of) their center(s) of mass, is chosen to act as the base point at which the tangent space is attached. The remaining sampled locations and sampled derivatives are mapped to said tangent space. This requires a vector transport between different tangent spaces. The actual interpolation is then conducted via classical vector space operations. The interpolant depends on the selected base point. The validity and performance of both approaches is illustrated by means of numerical examples.