Data Fitting on Manifolds with Composite Bézier-Like Curves and Blended Cubic Splines

  • Pierre-Yves Gousenbourger
  • Estelle Massart
  • P.-A. Absil


We propose several methods that address the problem of fitting a \(C^1\) curve \(\gamma \) to time-labeled data points on a manifold. The methods have a parameter, \(\lambda \), to adjust the relative importance of the two goals that the curve should meet: being “straight enough” while fitting the data “closely enough.” The methods are designed for ease of use: they only require to compute Riemannian exponentials and logarithms, they represent the curve by means of a number of tangent vectors that grows linearly with the number of data points, and, once the representation is computed, evaluating \(\gamma (t)\) at any t requires a small number of exponentials and logarithms (independent of the number of data points). Among the proposed methods, the blended cubic spline technique combines the additional properties of interpolating the data when \(\lambda \rightarrow \infty \) and reducing to the well-known cubic smoothing spline when the manifold is Euclidean. The methods are illustrated on synthetic and real data.


Riemannian manifold Data fitting Curve fitting Interpolation Smoothing Blended cubic spline 



The codes developed for this paper use the Manopt toolbox [7]. Special thanks to Benedikt Wirth for the very productive discussions: several methods presented here were conceived during an ongoing joint project on Bézier fitting surfaces.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Pierre-Yves Gousenbourger
    • 1
  • Estelle Massart
    • 2
  • P.-A. Absil
    • 2
  1. 1.ICTEAMUniversité catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.ICTEAMUniversité catholique de LouvainLouvain-la-NeuveBelgium

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