Document Open Access Logo

A Sparse Delaunay Filtration

Author Donald R. Sheehy

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2021.58.pdf
  • Filesize: 1.41 MB
  • 16 pages

Document Identifiers

Author Details

Donald R. Sheehy
  • North Carolina State University, Raleigh, NC, USA

Funding

This research was supported by the NSF under grant CCF-2017980.

Cite As Get BibTex

Donald R. Sheehy. A Sparse Delaunay Filtration. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 58:1-58:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.SoCG.2021.58

Abstract

We show how a filtration of Delaunay complexes can be used to approximate the persistence diagram of the distance to a point set in ℝ^d. Whereas the full Delaunay complex can be used to compute this persistence diagram exactly, it may have size O(n^⌈d/2⌉). In contrast, our construction uses only O(n) simplices. The central idea is to connect Delaunay complexes on progressively denser subsamples by considering the flips in an incremental construction as simplices in d+1 dimensions. This approach leads to a very simple and straightforward proof of correctness in geometric terms, because the final filtration is dual to a (d+1)-dimensional Voronoi construction similar to the standard Delaunay filtration. We also, show how this complex can be efficiently constructed.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Delaunay Triangulation
  • Persistent Homology
  • Sparse Filtrations

Metrics

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

Related Versions

References

  1. Franz Aurenhammer, Bert Jüttler, and Günter Paulini. Voronoi diagrams for parallel halflines and line segments in space. In Yoshio Okamoto and Takeshi Tokuyama, editors, 28th International Symposium on Algorithms and Computation (ISAAC 2017), volume 92 of Leibniz International Proceedings in Informatics (LIPIcs), pages 7:1-7:10. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  2. Magnus Bakke Botnan and Gard Spreemann. Approximating persistent homology in Euclidean space through collapses. Applicable Algebra in Engineering, Communication and Computing, 26(1):73-101, 2015. URL: http://arxiv.org/abs/1403.0533.
  3. A. Bowyer. Computing dirichlet tessellations. The Computer Journal, 2(24):162-166, 1981. Google Scholar
  4. Kevin Q. Brown. Voronoi diagrams from convex hulls. Information Processing Letters, 9(5):223-228, 1979. Google Scholar
  5. Mickaël Buchet, Frédéric Chazal, Steve Y. Oudot, and Donald R. Sheehy. Efficient and robust persistent homology for measures. In ACM-SIAM Symposium on Discrete Algorithms, pages 168-180, 2015. Google Scholar
  6. Nicholas J. Cavanna, Mahmoodreza Jahanseir, and Donald R. Sheehy. A geometric perspective on sparse filtrations. In Proceedings of the Canadian Conference on Computational Geometry, 2015. Google Scholar
  7. Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. The Structure and Stability of Persistence Modules. SpringerBriefs in Mathematics. Springer International Publishing, 2016. Google Scholar
  8. Frédéric Chazal and Steve Y. Oudot. Towards persistence-based reconstruction in Euclidean spaces. In Proceedings of the 24th ACM Symposium on Computational Geometry, pages 232-241, 2008. Google Scholar
  9. Siu-Wing Cheng, Tamal K. Dey, and Jonathan Richard Shewchuk. Delaunay Mesh Generation. CRC Press, 2012. Google Scholar
  10. L. Paul Chew. Guaranteed-Quality Mesh Generation for Curved Surfaces. In Proceedings of the Ninth Annual Symposium on Computational Geometry, pages 274-280, 1993. Google Scholar
  11. Aruni Choudhary, Michael Kerber, and Sharath Raghvendra. Improved topological approximations by digitization. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2675-2688, 2019. Google Scholar
  12. Kenneth L. Clarkson. Nearest neighbor searching in metric spaces: Experimental results for `sb(s)`. Preliminary version presented at ALENEX99, 2003. Google Scholar
  13. Tamal K. Dey, Dayu Shi, and Yusu Wang. Simba: An efficient tool for approximating rips-filtration persistence via simplicial batch-collapse. In 24th Annual European Symposium on Algorithms, pages 206:1-206:16, 2016. Google Scholar
  14. M.E. Dyer and A.M. Frieze. A simple heuristic for the p-centre problem. Operations Research Letters, 3(6):285-288, 1985. Google Scholar
  15. Herbert Edelsbrunner. The union of balls and its dual shape. Discrete & Computational Geometry, 13:415-440, 1995. Google Scholar
  16. Herbert Edelsbrunner. Geometry and Topology for Mesh Generation. Cambridge University Press, 2001. Google Scholar
  17. Herbert Edelsbrunner, David G. Kirkpatrick, and Raimund Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, 29(4):551-559, 1983. Google Scholar
  18. Herbert Edelsbrunner, David Letscher, and Afra Zomorodian. Topological persistence and simplification. Discrete & Computational Geometry, 4(28):511-533, 2002. Google Scholar
  19. Herbert Edelsbrunner and Raimund Seidel. Voronoi diagrams and arrangements. Discrete & Computational Geometry, 1(1):25-44, 1986. Google Scholar
  20. Herbert Edelsbrunner and Nimish R. Shah. Incremental topological flipping works for regular triangulations. Algorithmica, 15, 1996. Google Scholar
  21. Teofilo F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci., 38:293-306, 1985. Google Scholar
  22. Leonidas J Guibas. Kinetic data structures - a state of the art report. In Proc. Workshop Algorithmic Found. Robot, pages 191-209, 1998. Google Scholar
  23. Sariel Har-Peled and Manor Mendel. Fast construction of nets in low dimensional metrics, and their applications. SIAM Journal on Computing, 35(5):1148-1184, 2006. Google Scholar
  24. Benoît Hudson, Gary Miller, and Todd Phillips. Sparse Voronoi Refinement. In Proceedings of the 15th International Meshing Roundtable, pages 339-356, Birmingham, Alabama, 2006. Long version available as Carnegie Mellon University Technical Report CMU-CS-06-132. Google Scholar
  25. Benoît Hudson, Gary L. Miller, Steve Y. Oudot, and Donald R. Sheehy. Topological inference via meshing. In Proceedings of the 26th ACM Symposium on Computational Geometry, pages 277-286, 2010. Google Scholar
  26. Barry Joe. Three-dimensional triangulations from local transformations. SIAM J. Sci. Stat. Comput., 10:718-741, 1989. Google Scholar
  27. C. L. Lawson. Software for C1 surface interpolation. In J. R. Rice, editor, Mathematical Software, volume III, pages 161-194. Academic, New York, 1977. Google Scholar
  28. Charles L. Lawson. Transforming triangulations. Discrete Mathematics, 3:365-372, 1972. Google Scholar
  29. Gary L. Miller, Todd Phillips, and Donald R. Sheehy. Beating the spread: Time-optimal point meshing. In Proceedings of the 26th ACM Symposium on Computational Geometry, pages 321-330, 2011. Google Scholar
  30. Gary L. Miller and Donald R. Sheehy. A new approach to output-sensitive construction of voronoi diagrams and delaunay triangulations. Discrete & Computational Geometry, 52(3):476-491, 2014. URL: https://doi.org/10.1007/s00454-014-9629-y.
  31. Jim Ruppert. A Delaunay refinement algorithm for quality 2-dimensional mesh generation. J. Algorithms, 18(3):548-585, 1995. Google Scholar
  32. Raimund Seidel. On the number of faces in higher-dimensional Voronoi diagrams. In Proceedings of the 3rd Annual Symposium on Computational Geometry, pages 181-185, 1987. Google Scholar
  33. Donald R. Sheehy. Mesh Generation and Geometric Persistent Homology. PhD thesis, Carnegie Mellon University, 2011. CMU CS Tech Report CMU-CS-11-121. Google Scholar
  34. Donald R. Sheehy. New Bounds on the Size of Optimal Meshes. Computer Graphics Forum, 31(5):1627-1635, 2012. Google Scholar
  35. Donald R. Sheehy. Linear-size approximations to the Vietoris-Rips filtration. Discrete & Computational Geometry, 49(4):778-796, 2013. Google Scholar
  36. Donald R. Sheehy. An output-sensitive algorithm for computing weighted α-complexes. In Proceedings of the Canadian Conference on Computational Geometry, 2015. Google Scholar
  37. D. F. Watson. Computing the n-dimensional delaunay tessellation with application to voronoi polytopes. The Computer Journal, 24(2):167-172, 1981. 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