A Variational Model for Data Fitting on Manifolds by Minimizing the Acceleration of a Bézier Curve

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...

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Bibliographic Details
Published inFrontiers in applied mathematics and statistics Vol. 4
Main Authors Bergmann, Ronny, Gousenbourger, Pierre-Yves
Format Journal Article
LanguageEnglish
Published Frontiers Media S.A 12.12.2018
Subjects
composite Bézier curves
curve fitting
Jacobi fields
Riemannian manifolds
variational models
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Summary: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 capabilities and validity of this approach both for interpolation and approximation. The examples also illustrate that the approach outperforms previous works tackling this problem.AMS subject classification (2010). 65K10, 65D10, 65D25, 53C22, 49Q99.
ISSN:2297-4687
2297-4687
DOI:10.3389/fams.2018.00059