Encyclopedia of Algorithms

2016 Edition
| Editors: Ming-Yang Kao

Manifold Reconstruction

  • Siu-Wing ChengEmail author
Reference work entry
DOI: https://doi.org/10.1007/978-1-4939-2864-4_714

Years and Authors of Summarized Original Work

  • 2005; Cheng, Dey, Ramos

  • 2008; Niyogi, Smale, Weinberger

  • 2014; Boissonnat, Ghosh

  • 2014; Cheng, Chiu

Problem Definition

With the widespread of sensing and Internet technologies, a large number of numeric attributes for a physical or cyber phenomenon can now be collected. If each attribute is viewed as a coordinate, an instance in the collection can be viewed as a point in \(\mathbb{R}^{d}\)

Keywords

Čzech complex Delaunay complex Homology Homeomorphism Implicit function Voronoi diagram 
This is a preview of subscription content, log in to check access.

Recommended Reading

  1. 1.
    Attali D, Lieutier A, Salinas D (2013) Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes. Comput Geom Theory Appl 46(4):448–465MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Boissonnat J-D, Ghost A (2014) Manifold reconstruction using tangential Delaunay complexes. Discret Comput Geom 51(1):221–267MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Boissonnat J-D, Guibas LJ, Oudot S (2009) Manifold reconstruction in arbitrary dimensions using witness complexes. Discret Comput Geom 42(1):37–70MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cheng S-W, Chiu, M-K (2009) Dimension detection via slivers. In: Proceedings of the ACM-SIAM symposium on discrete algorithms, New York, pp 1001–1010Google Scholar
  5. 5.
    Cheng S-W, Chiu M-K (2014) Implicit manifold reconstruction. In: Proceedings of the ACM-SIAM symposium on discrete algorithms, Portland, pp 161–173Google Scholar
  6. 6.
    Cheng S-W, Dey TK, Edelsbrunner H, Facello MA, Teng S-H (2000) Sliver exudation. J ACM 17(1–2):51–68MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cheng S-W, Dey TK, Ramos EA (2005) Manifold reconstruction from point samples. In: Proceedings of the ACM-SIAM symposium on discrete algorithms, Vancouver, pp 1018–1027zbMATHGoogle Scholar
  8. 8.
    Cheng S-W, Wang Y, Wu Z (2008) Provable dimension detection using principal component analysis. Int J Comput Geom Appl 18(5):415–440MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Chiu M-K (2013) Manifold reconstruction from discrete point sets. PhD dissertation, Hong Kong University of Science and TechnologyCrossRefGoogle Scholar
  10. 10.
    Dey TK, Giesen J, Goswami S, Zhao W (2003) Shape dimension and approximation from samples. Discret Comput Geom 29(3):419–343MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Friedman J (1998) Computing Betti numbers via combinatorial laplacians. Algorithmica 21(4):331–346MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Niyogi P, Smale S, Weinberger S (2008) Finding the homology of submanifolds with high confidence from random samples. Discret Comput Geom 39(1):419–441MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringHong Kong University of Science and TechnologyHong KongChina