Blended smoothing splines on Riemannian manifolds

TL;DR

This paper introduces a method for fitting smooth curves on Riemannian manifolds by blending Euclidean Bézier curves in tangent spaces, ensuring smoothness and natural cubic spline properties, demonstrated on the sphere S2.

Contribution

The paper proposes a novel blending approach for smoothing splines on Riemannian manifolds that guarantees smoothness and reduces to classical splines in Euclidean space.

Findings

Method produces C1 smooth curves on manifolds.

Ensures the curve is a natural cubic smoothing spline in Euclidean space.

Demonstrated successfully on the sphere S2.

Abstract

We present a method to compute a fitting curve B to a set of data points d0,...,dm lying on a manifold M. That curve is obtained by blending together Euclidean B\'ezier curves obtained on different tangent spaces. The method guarantees several properties among which B is C1 and is the natural cubic smoothing spline when M is the Euclidean space. We show examples on the sphere S2 as a proof of concept.