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Corrected Curvature Measures

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Abstract

This paper proposes a new mathematical and computational tool for inferring the geometry of shapes known only through approximations such as triangulated or digital surfaces. The main idea is to decouple the position of the shape boundary from its normal vector field. To do so, we extend a classical tool of geometric measure theory, the normal cycle, so that it takes as input not only a surface but also a normal vector field. We formalize it as a current in the oriented Grassmann bundle \(\mathbb {R}^3 \times \mathbb {S}^2\). By choosing adequate differential forms, we define geometric measures like area, mean and Gaussian curvatures. We then show the stability of these measures when both position and normal input data are approximations of the underlying continuous shape. As a byproduct, our tool is able to correctly estimate curvatures over polyhedral approximations of shapes with explicit bounds, even when their natural normal are not correct, as long as an external convergent normal vector field is provided. Finally, the accuracy, convergence and stability under noise perturbation is evaluated experimentally onto digital surfaces.

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Notes

  1. See http://mathoverflow.net/questions/263452.

  2. So that \(\mathbf {e}_{ji}(p)=- \mathbf {e}_{ij}(p)\).

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Authors and Affiliations

  1. Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, Chambéry, France

    Jacques-Olivier Lachaud

  2. LAMA, Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, 77447, Marne-la-Vallée, France

    Pascal Romon

  3. Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, LJK, 38000, Grenoble, France

    Boris Thibert

Authors
  1. Jacques-Olivier Lachaud
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  2. Pascal Romon
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  3. Boris Thibert
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Correspondence to Pascal Romon.

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This work has been partly funded by CoMeDiC ANR-15-CE40-0006 research grant. The authors thank the anonymous reviewer for his thoughtful comments and suggestions.

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Lachaud, JO., Romon, P. & Thibert, B. Corrected Curvature Measures. Discrete Comput Geom 68, 477–524 (2022). https://doi.org/10.1007/s00454-022-00399-4

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  • DOI: https://doi.org/10.1007/s00454-022-00399-4

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