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Tighter Bounds for Reconstruction from ε-Samples

Author Håvard Bakke Bjerkevik

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Håvard Bakke Bjerkevik
  • Institute of Geometry, Technische Universität Graz, Austria

Funding

The author is supported by the Austrian Science Fund (FWF) grant number P 33765-N.

Acknowledgements

The author would like to thank Michael Kerber for insights into the size of Delaunay triangulations, and Stefan Ohrhallinger and Scott A. Mitchell for answering my questions about the state of the art of curve and surface reconstruction.

Cite AsGet BibTex

Håvard Bakke Bjerkevik. Tighter Bounds for Reconstruction from ε-Samples. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 9:1-9:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.9

Abstract

We show that reconstructing a curve in ℝ^d for d ≥ 2 from a 0.66-sample is always possible using an algorithm similar to the classical NN-Crust algorithm. Previously, this was only known to be possible for 0.47-samples in ℝ² and 1/3-samples in ℝ^d for d ≥ 3. In addition, we show that there is not always a unique way to reconstruct a curve from a 0.72-sample; this was previously only known for 1-samples. We also extend this non-uniqueness result to hypersurfaces in all higher dimensions.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
Keywords
  • Curve reconstruction
  • surface reconstruction
  • ε-sampling

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References

  1. Ahmed Abdelkader, Chandrajit L. Bajaj, Mohamed S. Ebeida, Ahmed H. Mahmoud, Scott A. Mitchell, John D. Owens, and Ahmad A. Rushdi. Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm. In 34th International Symposium on Computational Geometry (SoCG 2018), pages 1:1-1:16, 2018. Google Scholar
  2. Ernst Althaus and Kurt Mehlhorn. Traveling salesman-based curve reconstruction in polynomial time. SIAM Journal on Computing, 31(1):27-66, 2001. Google Scholar
  3. Nina Amenta, Marshall Bern, and David Eppstein. The crust and the β-skeleton: Combinatorial curve reconstruction. Graphical models and image processing, 60(2):125-135, 1998. Google Scholar
  4. Nina Amenta, Sunghee Choi, Tamal K. Dey, and Naveen Leekha. A simple algorithm for homeomorphic surface reconstruction. In Proceedings of the sixteenth annual symposium on Computational geometry (SCG 2000), pages 213-222, 2000. Google Scholar
  5. Dominique Attali. r-regular shape reconstruction from unorganized points. Computational Geometry, 10(4):239-247, 1998. Google Scholar
  6. Matthew Berger, Andrea Tagliasacchi, Lee M. Seversky, Pierre Alliez, Gael Guennebaud, Joshua A. Levine, Andrei Sharf, and Claudio T. Silva. A survey of surface reconstruction from point clouds. Computer Graphics Forum, 36(1):301-329, 2017. Google Scholar
  7. Fausto Bernardini and Chandrajit L. Bajaj. Sampling and reconstructing manifolds using alpha-shapes. In Proceedings of the 9th Canadian Conference on Computational Geometry (CCCG 1997), pages 193-198, 1997. Google Scholar
  8. Håvard Bakke Bjerkevik. Tighter bounds for reconstruction from ε-samples. arXiv preprint v2, 2022. URL: http://arxiv.org/abs/2112.03656.
  9. Harry Blum. A transformation for extracting new descriptors of shape. In Models for the Perception of Speech and Visual Form, pages 362-380. MIT Press, Cambridge, 1967. Google Scholar
  10. Frédéric Chazal and André Lieutier. Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees. Computational Geometry, 40(2):156-170, 2008. Google Scholar
  11. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational Geometry: Algorithms and Applications. Springer, 3rd edition, 2008. Google Scholar
  12. Luiz Henrique De Figueiredo and Jonas de Miranda Gomes. Computational morphology of curves. The Visual Computer, 11(2):105-112, 1994. Google Scholar
  13. Tamal K. Dey. Curve and Surface Reconstruction: Algorithms with Mathematical Analysis, volume 23. Cambridge University Press, 2006. Google Scholar
  14. Tamal K. Dey, Joachim Giesen, Edgar A. Ramos, and Bardia Sadri. Critical points of distance to an ε-sampling of a surface and flow-complex-based surface reconstruction. International Journal of Computational Geometry & Applications, 18(1-2):29-61, 2008. Google Scholar
  15. Tamal K. Dey and Piyush Kumar. A simple provable algorithm for curve reconstruction. In Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1999), volume 99, pages 893-894, 1999. Google Scholar
  16. Tamal K. Dey, Kurt Mehlhorn, and Edgar A. Ramos. Curve reconstruction: Connecting dots with good reason. Computational Geometry, 15(4):229-244, 2000. Google Scholar
  17. Tamal K. Dey and Rephael Wenger. Fast reconstruction of curves with sharp corners. International Journal of Computational Geometry & Applications, 12(05):353-400, 2002. Google Scholar
  18. Christopher Gold. Crust and anti-crust: a one-step boundary and skeleton extraction algorithm. In Proceedings of the fifteenth annual symposium on Computational geometry (SCG 1999), pages 189-196, 1999. Google Scholar
  19. Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Toth. Handbook of Discrete and Computational Geometry. CRC press, 3rd edition, 2017. Google Scholar
  20. Tobias Lenz. How to sample and reconstruct curves with unusual features. In EWCG: Proc. of the 22nd European Workshop on Computational Geometry, pages 29-32. Citeseer, 2006. Google Scholar
  21. Partha Niyogi, Stephen Smale, and Shmuel Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete & Computational Geometry, 39(1-3):419-441, 2008. Google Scholar
  22. Stefan Ohrhallinger, Scott A. Mitchell, and Michael Wimmer. Curve reconstruction with many fewer samples. Computer Graphics Forum, 35(5):167-176, 2016. Google Scholar
  23. Stefan Ohrhallinger, Jiju Peethambaran, Amal D. Parakkat, Tamal K. Dey, and Ramanathan Muthuganapathy. 2d points curve reconstruction survey and benchmark. Computer Graphics Forum, 40(2):611-632, 2021. Google Scholar
  24. Peer Stelldinger. Topologically correct surface reconstruction using alpha shapes and relations to ball-pivoting. In 19th International Conference on Pattern Recognition (ICPR 2008), pages 1-4, 2008. Google Scholar
  25. Peer Stelldinger and Leonid Tcherniavski. Provably correct reconstruction of surfaces from sparse noisy samples. Pattern Recognition, 42(8):1650-1659, 2009. Google Scholar
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