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C1 interpolating Bézier path on Riemannian manifolds, with applications to 3D shape space

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  • Samir, Chafik
  • Adouani, Ines

Abstract

This paper introduces a new framework to fit a C1 Bézier path to a given finite set of ordered data points on shape space of curves. We prove existence and uniqueness properties of the path and give a numerical method for constructing an optimal solution. Furthermore, we present a conceptually simple method to compute the optimal intermediate control points that define this path. The main property of the method is that when the manifold reduces to a Euclidean space or the finite dimensional sphere, the control points minimize the mean square acceleration. Potential applications of fitting smooth paths on Riemannian manifold include applications in robotics, animations, graphics, and medical studies. In this paper, we will focus on different medical applications to predict missing data from few ordered observations.

Suggested Citation

  • Samir, Chafik & Adouani, Ines, 2019. "C1 interpolating Bézier path on Riemannian manifolds, with applications to 3D shape space," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 371-384.
    • Handle: RePEc:eee:apmaco:v:348:y:2019:i:c:p:371-384
    DOI: 10.1016/j.amc.2018.11.060
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    References listed on IDEAS

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    1. Kim Kenobi & Ian L. Dryden & Huiling Le, 2010. "Shape curves and geodesic modelling," Biometrika, Biometrika Trust, vol. 97(3), pages 567-584.
    2. Peter E. Jupp & John T. Kent, 1987. "Fitting Smooth Paths to Spherical Data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 36(1), pages 34-46, March.
    3. Li, Xin & Chang, Yubo, 2018. "Non-uniform interpolatory subdivision surface," Applied Mathematics and Computation, Elsevier, vol. 324(C), pages 239-253.
    4. Allasia, Giampietro & Cavoretto, Roberto & De Rossi, Alessandra, 2018. "Hermite–Birkhoff interpolation on scattered data on the sphere and other manifolds," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 35-50.
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