Discrete & Computational Geometry

, Volume 47, Issue 2, pp 347–377

Optimal Topological Simplification of Discrete Functions on Surfaces

Open Access
Article

Abstract

Given a function f on a surface and a tolerance δ>0, we construct a function fδ subject to ‖fδfδ such that fδ has a minimum number of critical points. Our construction relies on a connection between discrete Morse theory and persistent homology and completely removes homological noise with persistence ≤2δ from the input function f. The number of critical points of the resulting simplified function fδ achieves the lower bound dictated by the stability theorem of persistent homology. We show that the simplified function can be computed in linear time after persistence pairs have been computed.

Keywords

Discrete Morse theory Persistent homology Topological denoising 

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Institute for Numerical and Applied MathematicsUniversity of GöttingenGöttingenGermany
  2. 2.Department of Mathematics and Computer ScienceFreie Universität BerlinBerlinGermany

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