A variational model for data fitting on manifolds by minimizing the acceleration of a B\'ezier curve

TL;DR

This paper introduces a variational model for fitting Bzier curves on Riemannian manifolds by minimizing acceleration, providing an efficient algorithm that outperforms previous methods for data interpolation and approximation.

Contribution

It develops a novel variational framework and a stable algorithm for fitting Bzier curves on manifolds, minimizing acceleration and improving over prior approaches.

Findings

The method effectively fits curves to data on manifolds.

The algorithm is numerically stable and efficient.

Results outperform previous methods in accuracy and stability.

Abstract

We derive a variational model to fit a composite B\'ezier curve to a set of data points on a Riemannian manifold. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a B\'ezier curve on manifolds with respect to its control points, expressed as a concatenation of so-called adjoint Jacobi fields. Several examples illustrate the capabilites and validity of this approach both for interpolation and approximation. The examples also illustrate that the approach outperforms previous works tackling this problem.