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Discrete & Computational Geometry

, Volume 57, Issue 3, pp 545–570 | Cite as

Voronoi-Based Estimation of Minkowski Tensors from Finite Point Samples

  • Daniel Hug
  • Markus Kiderlen
  • Anne Marie Svane
Article

Abstract

Intrinsic volumes and Minkowski tensors have been used to describe the geometry of real world objects. This paper presents an estimator that allows approximation of these quantities from digital images. It is based on a generalized Steiner formula for Minkowski tensors of sets of positive reach. When the resolution goes to infinity, the estimator converges to the true value if the underlying object is a set of positive reach. The underlying algorithm is based on a simple expression in terms of the cells of a Voronoi decomposition associated with the image.

Keywords

Minkowski tensor Digital algorithm Set of positive reach Digitization 

Mathematics Subject Classification

52A38 28A75 52A20 94A08 68U10 62H35 

Notes

Acknowledgements

We wish to thank the referees for carefully reading the paper and making helpful suggestions for improvements. The first author was supported in part by DFG grants FOR 1548 and HU 1874/4-2. The third author was supported by a grant from the Carlsberg Foundation. The second and third authors were supported by the Centre for Stochastic Geometry and Advanced Bioimaging, funded by the Villum Foundation.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Daniel Hug
    • 1
  • Markus Kiderlen
    • 2
  • Anne Marie Svane
    • 2
  1. 1.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.Department of MathematicsAarhus UniversityAarhus CDenmark

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