Skip to main content

Interleaved Filtrations: Theory and Applications in Point Cloud Data Analysis

  • Conference paper
Geometric Science of Information (GSI 2013)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 8085))

Included in the following conference series:

Abstract

The aim of the talk at GSI’2013 is to introduce the audience to the concept of interleaving between filtrations, coming from applied topology, and to emphasize its influence on emerging trends in point cloud data analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Biasotti, S., De Floriani, L., Falcidieno, B., Frosini, P., Giorgi, D., Landi, C., Papaleo, L., Spagnuolo, M.: Describing shapes by geometrical-topological properties of real functions. ACM Comput. Surv. 40(4), 12:1–12:87 (2008)

    Google Scholar 

  2. Boissonnat, J.-D., Guibas, L.J., Oudot, S.Y.: Manifold reconstruction in arbitrary dimensions using witness complexes. In: Proc. 23rd Sympos. on Comp. Geom., pp. 194–203 (2007)

    Google Scholar 

  3. Carlsson, G., de Silva, V.: Zigzag persistence. Foundations of Computational Mathematics 10(4), 367–405 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.Y.: Proximity of persistence modules and their diagrams. In: Proc. 25th Annu. Symposium on Computational Geometry, pp. 237–246 (2009)

    Google Scholar 

  5. Chazal, F., Cohen-Steiner, D., Lieutier, A.: A sampling theory for compact sets in euclidean spaces. Discrete Comput. Geom. 41(3) (2009)

    Google Scholar 

  6. Chazal, F., de Silva, V., Glisse, M., Oudot, S.Y.: The structure and stability of persistence modules. Research Report arXiv:1207.3674 [math.AT] (July 2012)

    Google Scholar 

  7. Chazal, F., de Silva, V., Oudot, S.Y.: Persistence stability for geometric complexes. Research Report arXiv:1207.3885 [math.AT] (July 2012)

    Google Scholar 

  8. Chazal, F., Guibas, L.J., Oudot, S.Y., Skraba, P.: Analysis of scalar fields over point cloud data. In: Proc. 19th ACM-SIAM Sympos. on Discrete Algorithms, pp. 1021–1030 (2009)

    Google Scholar 

  9. Chazal, F., Guibas, L.J., Oudot, S.Y., Skraba, P.: Persistence-based clustering in riemannian manifolds. In: Proc. 27th Annu. ACM Sympos. on Comput. Geom., pp. 97–106 (June 2011)

    Google Scholar 

  10. Chazal, F., Lieutier, A.: Stability and computation of topological invariants of solids in ℝn. Discrete Comput. Geom. 37(4), 601–617 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chazal, F., Oudot, S.Y.: Towards persistence-based reconstruction in Euclidean spaces. In: Proc. 24th ACM Sympos. Comput. Geom., pp. 232–241 (2008)

    Google Scholar 

  12. Chazal, F., Cohen-Steiner, D., Guibas, L.J., Mémoli, F., Oudot, S.: Gromov-hausdorff stable signatures for shapes using persistence. Comput. Graph. Forum 28(5), 1393–1403 (2009)

    Article  Google Scholar 

  13. Cheng, S.-W., Dey, T.K., Ramos, E.A.: Manifold reconstruction from point samples. In: Proc. 16th Sympos. Discrete Algorithms, pp. 1018–1027 (2005)

    Google Scholar 

  14. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete & Computational Geometry 37(1), 103–120 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. de Silva, V.: A weak characterisation of the Delaunay triangulation. Geometriae Dedicata 135(1), 39–64 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. de Silva, V., Carlsson, G.: Topological estimation using witness complexes. In: Proc. Sympos. Point-Based Graphics, pp. 157–166 (2004)

    Google Scholar 

  17. de Silva, V., Ghrist, R.: Homological sensor networks. Notices of the American Mathematical Society 54(1), 10–17 (2007)

    MathSciNet  MATH  Google Scholar 

  18. Dey, T.K., Fan, F., Wang, Y.: Computing topological persistence for simplicial maps. Research Report arXiv:1208.5018 [cs.CG] (August 2012)

    Google Scholar 

  19. Dionysus. By Dmitriy Morozov, http://www.mrzv.org/software/dionysus/

  20. Edelsbrunner, H., Harer, J.: Persistent homology—a survey. In: Goodman, J.E., Pach, J., Pollack, R. (eds.) Surveys on Discrete and Computational Geometry. Twenty Years Later, pp. 257–282. Amer. Math. Soc. (2008)

    Google Scholar 

  21. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Proc. 41st Annu. IEEE Sympos. Found. Comput. Sci., pp. 454–463 (2000)

    Google Scholar 

  22. Hudson, B., Miller, G.L., Oudot, S.Y., Sheehy, D.R.: Topological inference via meshing. In: Proc. 26th Annual Symposium on Computational Geometry, pp. 277–286 (2010)

    Google Scholar 

  23. Lee, A.B., Pederson, K.S., Mumford, D.: The nonlinear statistics of high-contrast patches in natural images. Internat. J. of Computer Vision 54(1-3), 83–103 (2003)

    Article  MATH  Google Scholar 

  24. Oudot, S.Y., Sheehy, D.R.: Zigzag zoology: Rips zigzags for homology inference. In: Proc. 29th Annual Symposium on Computational Geometry (to appear, 2013)

    Google Scholar 

  25. Sheehy, D.R.: Linear-size approximations to the vietoris-rips filtration. In: Proc. 28th Symposium on Computational Geometry, pp. 239–248 (2012)

    Google Scholar 

  26. Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Chazal, F., Oudot, S.Y. (2013). Interleaved Filtrations: Theory and Applications in Point Cloud Data Analysis. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2013. Lecture Notes in Computer Science, vol 8085. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40020-9_65

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-40020-9_65

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40019-3

  • Online ISBN: 978-3-642-40020-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics