Localized Homology

In this paper, we provide the theoretical foundation and an effective algorithm for localizing topological attributes such as tunnels and voids. Unlike previous work that focused on 2-manifolds with restricted geometry, our theory is general and localizes arbitrary-dimensional attributes in arbitrary spaces. We implement our algorithm and present experiments to validate our approach in practice.

[1]  Zoë J. Wood,et al.  Topological Noise Removal , 2001, Graphics Interface.

[2]  Francis Lazarus,et al.  Optimal System of Loops on an Orientable Surface , 2005, Discret. Comput. Geom..

[3]  Jeff Erickson,et al.  Optimally Cutting a Surface into a Disk , 2002, SCG '02.

[4]  Herbert Edelsbrunner,et al.  Computing linking numbers of a filtration. , 2003 .

[6]  John R. Harper,et al.  Algebraic topology : a first course , 1982 .

[7]  Daniel Freedman,et al.  Measuring and Localizing Homology Classes , 2008 .

[8]  Leonidas J. Guibas,et al.  Locating and bypassing routing holes in sensor networks , 2004, IEEE INFOCOM 2004.

[9]  Leonidas J. Guibas,et al.  Persistence barcodes for shapes , 2004, SGP '04.

[10]  Marc Levoy,et al.  The digital Michelangelo project: 3D scanning of large statues , 2000, SIGGRAPH.

[11]  Graeme Segal,et al.  Classifying spaces and spectral sequences , 1968 .

[12]  Francis Lazarus,et al.  Optimal System of Loops on an Orientable Surface , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[13]  Jon P. May.  Simplicial objects in algebraic topology , 1993 .

[14]  Vin de Silva,et al.  On the Local Behavior of Spaces of Natural Images , 2007, International Journal of Computer Vision.

[15]  Paul G. Goerss,et al.  Simplicial Homotopy Theory , 2009, Modern Birkhäuser Classics.

[16]  日本数学会,et al.  Encyclopedic dictionary of mathematics , 1993 .

[17]  James R. Munkres,et al.  Elements of algebraic topology , 1984 .

[18]  R. Ho.  Algebraic Topology , 2022 .

[19]  Jean-Claude Latombe,et al.  Stochastic roadmap simulation: an efficient representation and algorithm for analyzing molecular motion , 2002, RECOMB '02.

[20]  M. Hopkins.  Equivariant K-theory , 1968 .

[21]  Mathieu Desbrun,et al.  Removing excess topology from isosurfaces , 2004, TOGS.

[22]  Kenneth S. Brown,et al.  Cohomology of Groups , 1982 .

[23]  Edward B Curtis,et al.  Simplicial homotopy theory , 1968 .

[24]  Afra Zomorodian,et al.  Computing Persistent Homology , 2005, Discret. Comput. Geom..

[25]  Herbert Edelsbrunner,et al.  Topological Persistence and Simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[26]  Afra Zomorodian,et al.  Computing Persistent Homology , 2004, SCG '04.

[27]  Matthew H. Austern.  Generic programming and the STL - using and extending the C++ standard template library , 1999, Addison-Wesley professional computing series.

[28]  Herbert Edelsbrunner,et al.  Computing Linking Numbers of a Filtration , 2001, WABI.

[29]  Jon Louis Bentley,et al.  Multidimensional binary search trees used for associative searching , 1975, CACM.

[30]  Jeff Erickson,et al.  Greedy optimal homotopy and homology generators , 2005, SODA '05.

[31]  Anne Verroust-Blondet,et al.  Computing a canonical polygonal schema of an orientable triangulated surface , 2001, SCG '01.

[32]  Chee-Keng Yap,et al.  Computational complexity of combinatorial surfaces , 1990, SCG '90.

[33]  Afra J. Zomorodian,et al.  Topology for Computing (Cambridge Monographs on Applied and Computational Mathematics) , 2005 .

[34]  James R. Munkres,et al.  Topology; a first course , 1974 .