Document Open Access Logo

Delaunay-Like Triangulation of Smooth Orientable Submanifolds by 𝓁₁-Norm Minimization

Authors Dominique Attali, André Lieutier

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2022.8.pdf
  • Filesize: 0.81 MB
  • 16 pages

Document Identifiers

Author Details

Dominique Attali
  • Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, Grenoble, France
André Lieutier
  • Dassault systèmes, Aix-en-Provence, France

Acknowledgements

We are grateful to the anonymous referees for carefully reading the paper and many helpful suggestions.

Cite As Get BibTex

Dominique Attali and André Lieutier. Delaunay-Like Triangulation of Smooth Orientable Submanifolds by 𝓁₁-Norm Minimization. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SoCG.2022.8

Abstract

In this paper, we focus on one particular instance of the shape reconstruction problem, in which the shape we wish to reconstruct is an orientable smooth submanifold of the Euclidean space. Assuming we have as input a simplicial complex K that approximates the submanifold (such as the Čech complex or the Rips complex), we recast the reconstruction problem as a 𝓁₁-norm minimization problem in which the optimization variable is a chain of K. Providing that K satisfies certain reasonable conditions, we prove that the considered minimization problem has a unique solution which triangulates the submanifold and coincides with the flat Delaunay complex introduced and studied in a companion paper [D. Attali and A. Lieutier, 2022]. Since the objective is a weighted 𝓁₁-norm and the contraints are linear, the triangulation process can thus be implemented by linear programming.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • manifold reconstruction
  • Delaunay complex
  • triangulation
  • sampling conditions
  • optimization
  • 𝓁₁-norm minimization
  • simplicial complex
  • chain
  • fundamental class

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Pierre Alliez, David Cohen-Steiner, Mariette Yvinec, and Mathieu Desbrun. Variational tetrahedral meshing. ACM Transactions on Graphics (TOG), 24(3):617-625, 2005. Google Scholar
  2. D. Attali and A. Lieutier. Delaunay-like triangulation of smooth orientable submanifolds by 𝓁₁-norm minimization, 2022. URL: http://arxiv.org/abs/2203.06008.
  3. D. Attali and A. Lieutier. Flat delaunay complexes for homeomorphic manifold reconstruction, 2022. URL: http://arxiv.org/abs/2203.05943.
  4. D. Attali, A. Lieutier, and D. Salinas. Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes. Computational Geometry: Theory and Applications, 46(4):448-465, 2013. Google Scholar
  5. J.-D. Boissonnat and A. Ghosh. Manifold reconstruction using tangential delaunay complexes. Discrete & Computational Geometry, 51(1):221-267, 2014. Google Scholar
  6. Jean-Daniel Boissonnat, Frédéric Chazal, and Mariette Yvinec. Geometric and topological inference, volume 57. Cambridge University Press, 2018. Google Scholar
  7. Jean-Daniel Boissonnat, Ramsay Dyer, and Arijit Ghosh. The stability of delaunay triangulations. International Journal of Computational Geometry & Applications, 23(04n05):303-333, August 2013. URL: https://doi.org/10.1142/s0218195913600078.
  8. Jean-Daniel Boissonnat and Mariette Yvinec. Algorithmic geometry. Cambridge university press, 1998. Google Scholar
  9. Glencora Borradaile, William Maxwell, and Amir Nayyeri. Minimum Bounded Chains and Minimum Homologous Chains in Embedded Simplicial Complexes. In Sergio Cabello and Danny Z. Chen, editors, 36th International Symposium on Computational Geometry (SoCG 2020), volume 164 of Leibniz International Proceedings in Informatics (LIPIcs), pages 21:1-21:15, Dagstuhl, Germany, 2020. Schloss Dagstuhlendash Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.21.
  10. Erin W. Chambers, Jeff Erickson, and Amir Nayyeri. Minimum cuts and shortest homologous cycles. In Proceedings of the 25th Annual Symposium on Computational Geometry - SCG '09, page 377, Aarhus, Denmark, 2009. ACM Press. URL: https://doi.org/10.1145/1542362.1542426.
  11. F. Chazal, D. Cohen-Steiner, and A. Lieutier. A sampling theory for compact sets in Euclidean space. Discrete and Computational Geometry, 41(3):461-479, 2009. Google Scholar
  12. F. Chazal and A. Lieutier. Smooth Manifold Reconstruction from Noisy and Non Uniform Approximation with Guarantees. Computational Geometry: Theory and Applications, 40:156-170, 2008. Google Scholar
  13. Chao Chen and Daniel Freedman. Hardness Results for Homology Localization. Discrete & Computational Geometry, 45(3):425-448, April 2011. URL: https://doi.org/10.1007/s00454-010-9322-8.
  14. L. Chen and M. Holst. Efficient mesh optimization schemes based on optimal delaunay triangulations. Computer Methods in Applied Mechanics and Engineering, 200(9):967-984, 2011. Google Scholar
  15. Long Chen. Mesh smoothing schemes based on optimal delaunay triangulations. In Proceedings of the 13th International Meshing Roundtable, IMR 2004, Williamsburg, Virginia, USA, September 19-22, 2004, pages 109-120, 2004. URL: http://imr.sandia.gov/papers/abstracts/Ch317.html.
  16. Long Chen and Jin-chao Xu. Optimal delaunay triangulations. Journal of Computational Mathematics, pages 299-308, 2004. Google Scholar
  17. Zhonggui Chen, Wenping Wang, Bruno Lévy, Ligang Liu, and Feng Sun. Revisiting optimal delaunay triangulation for 3d graded mesh generation. SIAM Journal on Scientific Computing, 36(3):A930-A954, 2014. Google Scholar
  18. David Cohen-Steiner, André Lieutier, and Julien Vuillamy. Regular triangulations as lexicographic optimal chains. Preprint HAL, 2019. URL: https://hal.archives-ouvertes.fr/hal-02391285.
  19. Tamal K. Dey, Anil N. Hirani, and Bala Krishnamoorthy. Optimal Homologous Cycles, Total Unimodularity, and Linear Programming. SIAM Journal on Computing, 40(4):1026-1044, January 2011. URL: https://doi.org/10.1137/100800245.
  20. Tamal K. Dey, Tao Hou, and Sayan Mandal. Computing minimal persistent cycles: Polynomial and hard cases. In Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '20, pages 2587-2606, USA, January 2020. Society for Industrial and Applied Mathematics. Google Scholar
  21. H. Edelsbrunner. Geometry and topology for mesh generation. Cambridge Univ Pr, 2001. Google Scholar
  22. Herbert Edelsbrunner and Nimish R Shah. Incremental topological flipping works for regular triangulations. Algorithmica, 15(3):223-241, 1996. Google Scholar
  23. H. Federer. Curvature measures. Trans. Amer. Math. Soc, 93:418-491, 1959. Google Scholar
  24. Jisu Kim, Jaehyeok Shin, Frédéric Chazal, Alessandro Rinaldo, and Larry Wasserman. Homotopy reconstruction via the cech complex and the vietoris-rips complex. arXiv preprint, 2019. URL: http://arxiv.org/abs/1903.06955.
  25. Frank Morgan. Geometric measure theory: a beginner’s guide. Academic press, 2016. Google Scholar
  26. Robin A Moser and Gábor Tardos. A constructive proof of the general lovász local lemma. Journal of the ACM (JACM), 57(2):1-15, 2010. Google Scholar
  27. J.R. Munkres. Elements of algebraic topology. Perseus Books, 1993. Google Scholar
  28. Oleg R Musin. Properties of the delaunay triangulation. In Proceedings of the thirteenth annual symposium on Computational geometry, pages 424-426, 1997. Google Scholar
  29. P. Niyogi, S. Smale, and S. Weinberger. Finding the Homology of Submanifolds with High Confidence from Random Samples. Discrete Computational Geometry, 39(1-3):419-441, 2008. Google Scholar
  30. Samuel Rippa. Minimal roughness property of the delaunay triangulation. Computer Aided Geometric Design, 7(6):489-497, 1990. Google Scholar
  31. Julien Vuillamy. Simplification planimétrique et chaînes lexicographiques pour la reconstruction 3D de scènes urbaines. PhD thesis, Université Côte d'Azur, 2021. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact