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Front. Appl. Math. Stat. | doi: 10.3389/fams.2018.00059

A variational model for data fitting on manifolds by minimizing the acceleration of a Bézier curve

  • 1Technische Universität Chemnitz, Germany
  • 2Université Catholique de Louvain, Belgium

We derive a variational model to fit a composite Bézier 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 Bézier 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.

Keywords: Riemannian manifolds, Curve fitting, composite Bézier curves, Jacobi fields, variational models

Received: 26 Jul 2018; Accepted: 13 Nov 2018.

Edited by:

Luca Marchetti, COSBI - Center for Computational and Systems Biology, University of Trento, Italy

Reviewed by:

Vincenzo Bonnici, University of Verona, Italy
Giovanni Mascali, Università della Calabria, Italy  

Copyright: © 2018 Bergmann and Gousenbourger. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Dr. Ronny Bergmann, Technische Universität Chemnitz, Chemnitz, Germany, university@ronnybergmann.net