Canonical homotopy class representative using hyperbolic structure

Wei Zeng, Miao Jin, Feng Luo, Xianfeng David Gu

Research output: Chapter in Book/Report/Conference proceedingConference contribution

18 Scopus citations

Abstract

Homotopy group plays a role in computational topology with a fundamental importance. Each homotopy equivalence class contains an infinite number of loops. Finding a canonical representative within a homotopy class will simplify many computational tasks in computational topology, such as loop homotopy detection, pants decomposition. Furthermore, the canonical representative can be used as the shape descriptor. This work introduces a rigorous and practical method to compute a unique representative for each homotopy class. The main strategy is to use hyperbolic structure, such that each homotopy class has a unique closed geodesic, which is the representative. The following is the algorithm pipeline: for a given surface with negative Euler number, we apply hyperbolic Yamabe curvature flow to compute the unique Riemannian metric, which has constant negative one curvature everywhere and is conformal to the original metric. Then we compute the Fuchsian group generators of the surface on the hyperbolic space. For a given loop on the surface, we lift it to the universal covering space, to obtain the Fuchsian transformation corresponding to the homotopy class of the loop. The unique closed geodesic inside the homotopy class is the axis of the Fuchsian transformation, which is the canonical representative. Theories and algorithms are explained thoroughly in details. Experimental results are reported to show the efficiency and efficacy of the algorithm. The unique homotopy class representative can be applied for homotopy detection and shape comparison.

Original languageEnglish (US)
Title of host publication2009 IEEE International Conference on Shape Modeling and Applications, SMI 2009
Pages171-178
Number of pages8
DOIs
StatePublished - Nov 20 2009
Event2009 IEEE International Conference on Shape Modeling and Applications, SMI 2009 - Beijing, China
Duration: Jun 26 2009Jun 28 2009

Publication series

Name2009 IEEE International Conference on Shape Modeling and Applications, SMI 2009

Other

Other2009 IEEE International Conference on Shape Modeling and Applications, SMI 2009
CountryChina
CityBeijing
Period6/26/096/28/09

All Science Journal Classification (ASJC) codes

  • Computational Theory and Mathematics
  • Computer Science Applications
  • Computer Vision and Pattern Recognition

Keywords

  • Homotopy class
  • Hyperbolic Yamabe flow
  • Hyperbolic structure

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