Extending Persistence Using Poincare and Lefschetz Duality

被引：87

作者：

Cohen-Steiner, David ^{[1
]}

Edelsbrunner, Herbert ^{[2
,3
]}

Harer, John ^{[4
,5
]}

机构：

[1] INRIA, Sophia Antipolis, France

[2] Duke Univ, Dept Comp Sci, Durham, NC 27706 USA

[3] Geomagic, Res Triangle Pk, NC USA

[4] Duke Univ, Dept Math, Durham, NC 27706 USA

[5] Duke Univ, Ctr Computat Sci Engn & Med, Durham, NC 27706 USA

来源：

FOUNDATIONS OF COMPUTATIONAL MATHEMATICS | 2009年 / 9卷 / 01期

基金：

美国国家科学基金会;

关键词：

Computational topology; Homology; Persistence; Manifolds; Duality; Simplicial complexes; Algorithms;

D O I：

10.1007/s10208-008-9027-z

中图分类号：

TP301 [理论、方法];

学科分类号：

080201 [机械制造及其自动化];

摘要：

Persistent homology has proven to be a useful tool in a variety of contexts, including the recognition and measurement of shape characteristics of surfaces in 3. Persistence pairs homology classes that are born and die in a filtration of a topological space, but does not pair its actual homology classes. For the sublevelset filtration of a surface in 3, persistence has been extended to a pairing of essential classes using Reeb graphs. In this paper, we give an algebraic formulation that extends persistence to essential homology for any filtered space, present an algorithm to calculate it, and describe how it aids our ability to recognize shape features for codimension 1 submanifolds of Euclidean space. The extension derives from Poincaré duality but generalizes to nonmanifold spaces. We prove stability for general triangulated spaces and duality as well as symmetry for triangulated manifolds. © 2008 SFoCM.

引用

页码：79 / 103

页数：25

共 12 条

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Extreme elevation on a 2-manifold [J].