IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v348y2019icp371-384.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300318310348
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.amc.2018.11.060?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Mustafa, Ghulam & Hameed, Rabia, 2019. "Families of non-linear subdivision schemes for scattered data fitting and their non-tensor product extensions," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 214-240.
    2. Ian L. Dryden & Kwang-Rae Kim & Huiling Le, 2019. "Bayesian Linear Size-and-Shape Regression with Applications to Face Data," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 83-103, February.
    3. Samir, Chafik & Huang, Wen, 2021. "Coordinate descent optimization for one-to-one correspondence and supervised classification of 3D shapes," Applied Mathematics and Computation, Elsevier, vol. 388(C).
    4. Kwang‐Rae Kim & Ian L. Dryden & Huiling Le & Katie E. Severn, 2021. "Smoothing splines on Riemannian manifolds, with applications to 3D shape space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(1), pages 108-132, February.
    5. Hanna, Martin S. & Chang, Ted, 2000. "Fitting Smooth Histories to Rotation Data," Journal of Multivariate Analysis, Elsevier, vol. 75(1), pages 47-61, October.
    6. Xia, Peng & Lei, Na & Dong, Tian, 2023. "On the linearization methods for univariate Birkhoff rational interpolation," Applied Mathematics and Computation, Elsevier, vol. 445(C).
    7. García-Morales, Vladimir, 2021. "A constructive theory of shape," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    8. Beran, Rudolf, 2016. "Nonparametric estimation of trend in directional data," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3808-3827.
    9. Kaijun Peng & Jieqing Tan & Li Zhang, 2023. "Polynomial-Based Non-Uniform Ternary Interpolation Surface Subdivision on Quadrilateral Mesh," Mathematics, MDPI, vol. 11(2), pages 1-22, January.
    10. Di Noia, Antonio & Mastrantonio, Gianluca & Jona Lasinio, Giovanna, 2024. "Bayesian size-and-shape regression modelling," Statistics & Probability Letters, Elsevier, vol. 204(C).
    11. Meisam Moghimbeygi & Mousa Golalizadeh, 2019. "A longitudinal model for shapes through triangulation," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 103(1), pages 99-121, March.
    12. Baotao Chi & Shengmin Bai & Qianjian Guo & Yaoming Zhang & Wei Yuan & Can Li, 2023. "A New Definition of the Dual Interpolation Curve for CAD Modeling and Geometry Defeaturing," Mathematics, MDPI, vol. 11(16), pages 1-19, August.
    13. B. Cooper Boniece & Lajos Horv'ath & Lorenzo Trapani, 2023. "On changepoint detection in functional data using empirical energy distance," Papers 2310.04853, arXiv.org.
    14. Cavoretto, R. & De Rossi, A. & Perracchione, E., 2023. "Learning with Partition of Unity-based Kriging Estimators," Applied Mathematics and Computation, Elsevier, vol. 448(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:348:y:2019:i:c:p:371-384. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.