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On Multigrid Convergence of Local Algorithms for Intrinsic Volumes

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Abstract

Local digital algorithms based on n×⋯×n configuration counts are commonly used within science for estimating intrinsic volumes from binary images. This paper investigates multigrid convergence of such algorithms. It is shown that local algorithms for intrinsic volumes other than volume are not multigrid convergent on the class of convex polytopes. In fact, counter examples are plenty. On the other hand, for convex particles in 2D with a lower bound on the interior angles, a multigrid convergent local algorithm for the Euler characteristic is constructed. Also on the class of r-regular sets, counter examples to multigrid convergence are constructed for the surface area and the integrated mean curvature.

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Acknowledgements

The author is supported by Centre for Stochastic Geometry and Advanced Bioimaging, funded by the Villum Foundation. The author is most thankful to Markus Kiderlen for suggesting this problem in the first place and for useful input along the way.

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  1. Department of Mathematics, Aarhus University, 8000, Aarhus C, Denmark

    Anne Marie Svane

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  1. Anne Marie Svane
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Correspondence to Anne Marie Svane.

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Svane, A.M. On Multigrid Convergence of Local Algorithms for Intrinsic Volumes. J Math Imaging Vis 49, 148–172 (2014). https://doi.org/10.1007/s10851-013-0450-7

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  • DOI: https://doi.org/10.1007/s10851-013-0450-7

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