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Journal of Mathematical Imaging and Vision

, Volume 50, Issue 1–2, pp 164–177 | Cite as

Stable Length Estimates of Tube-Like Shapes

  • Herbert Edelsbrunner
  • Florian Pausinger
Article

Abstract

Motivated by applications in biology, we present an algorithm for estimating the length of tube-like shapes in 3-dimensional Euclidean space. In a first step, we combine the tube formula of Weyl with integral geometric methods to obtain an integral representation of the length, which we approximate using a variant of the Koksma–Hlawka Theorem. In a second step, we use tools from computational topology to decrease the dependence on small perturbations of the shape. We present computational experiments that shed light on the stability and the convergence rate of our algorithm.

Keywords

Tubes Length Integral geometry Quasi-Monte Carlo integration Quermassintegrals Discrepancy Persistent homology Algorithms Stability 

Notes

Acknowledgements

The authors thank Olga Symonova and Michael Kerber for sharing their implementation of the persistence algorithm. Furthermore, they thank three reviewers for their careful reading of two earlier manuscripts and for a number of insightful comments which helped improve the paper.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.IST Austria (Institute of Science and Technology Austria)KlosterneuburgAustria

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