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Shape Analysis by Computing Geodesics on a Manifold via Cubic B-splines

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Abstract

When a parameterized probability density function is used to represent a landmark-based shape, the shape can be viewed as a point on the manifold that equips with a Riemannian metric corresponding to the mixture models. Hence, given two shapes parameterized by the same density model, the geodesic distance between them can be used for an appropriate shape distance measure. We provide a computational strategy, which is based on the cubic B-splines, to get geodesics and geodesic distances between plane shapes represented by the mixture of Gaussians. In contrast to the methods that discretize geodesic into a sequence of line segments, the proposed method is computationally efficient and numerically stable.

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Notes

  1. http://vision.lems.brown.edu/content/available-software-and-databases#Datasets-Shape.

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Acknowledgements

We would like to thank the anonymous referees for providing us with constructive comments and suggestions.

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Authors and Affiliations

  1. School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing, 211816, People’s Republic of China

    Qian Ni

  2. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences (CAS), Beijing, 100190, People’s Republic of China

    Qian Ni

  3. Department of Mathematics, Hohai University, Nanjing, 211100, People’s Republic of China

    Xuhui Wang

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  1. Qian Ni
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Correspondence to Xuhui Wang.

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The authors declare no conflict of interest.

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This work is partially supported by grants from the National Natural Science Foundation of China (NSFC Nos. 12201292, 61772167, and 11771420), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (22KJB110015).

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Ni, Q., Wang, X. Shape Analysis by Computing Geodesics on a Manifold via Cubic B-splines. Commun. Math. Stat. (2023). https://doi.org/10.1007/s40304-023-00373-3

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  • DOI: https://doi.org/10.1007/s40304-023-00373-3

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