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Robust and Convergent Curvature and Normal Estimators with Digital Integral Invariants

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Modern Approaches to Discrete Curvature

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2184))

Abstract

We present, in details, a generic tool to estimate differential geometric quantities on digital shapes, which are subsets of \(\mathbb{Z}^{d}\). This tool, called digital integral invariant, simply places a ball at the point of interest, and then examines the intersection of this ball with input data to infer local geometric information. Just counting the number of input points within the intersection provides curvature estimation in 2D and mean curvature estimation in 3D. The covariance matrix of the same point set allows to recover principal curvatures, principal directions and normal direction estimates in 3D. We show the multigrid convergence of all these estimators, which means that their estimations tend toward the exact geometric quantities on—smooth enough—Euclidean shapes digitized with finer and finer gridsteps. During the course of the chapter, we establish several multigrid convergence results of digital volume and moments estimators in arbitrary dimensions. Afterwards, we show how to set automatically the radius parameter while keeping multigrid convergence properties. Our estimators are then demonstrated to be accurate in practice, with extensive comparisons with state-of-the-art methods. To conclude the exposition, we discuss their robustness to perturbations and noise in input data and we show how such estimators can detect features using scale-space arguments.

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Notes

  1. 1.

    ⊗ denotes the usual tensor product in vector spaces.

  2. 2.

    There is a typographic error in the λ1 expression in [ 48].

  3. 3.

    B R +(x) denotes the x-positive half ball of center x and radius R. Remember that x i is the i−th component of x.

  4. 4.

    The DGtal library is available at http://dgtal.org.

  5. 5.

    For an implementation, we refer to the CGAL library available at http://www.cgal.org.

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Acknowledgements

This work has been mainly funded by DigitalSnow ANR-11-BS02-009, KIDICO ANR-2010-BLAN-0205 and PRIMES Labex ANR-11-LABX-0063/ANR-11-IDEX-0007 research grants.

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Authors and Affiliations

  1. CNRS, LAMA, UMR 5127, Université Savoie Mont Blanc, F-73376, Le Bourget-du-lac, France

    Jacques-Olivier Lachaud & Jérémy Levallois

  2. CNRS, LJK, UMR 5224, Université Grenoble-Alpes, F-38041, Saint-Martin-d’Héres, France

    Jacques-Olivier Lachaud

  3. CNRS, Université de Lyon, LIRIS, UMR 5205, Université de Lyon, F-69621, Lyon, France

    David Coeurjolly

  4. CNRS, INSA-Lyon, LIRIS, UMR 5205, Université de Lyon, F-69621, Lyon, France

    Jérémy Levallois

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  1. Jacques-Olivier Lachaud
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  2. David Coeurjolly
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  3. Jérémy Levallois
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Correspondence to Jacques-Olivier Lachaud .

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Editors and Affiliations

  1. Laboratoire d’Informatique Gaspard Monge, Université Paris-Est - ESIEE Paris, Noisy-le-Grand, France

    Laurent Najman

  2. Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris-Est Marne-la-Vallée, Marne-la-Vallée, France

    Pascal Romon

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Lachaud, JO., Coeurjolly, D., Levallois, J. (2017). Robust and Convergent Curvature and Normal Estimators with Digital Integral Invariants. In: Najman, L., Romon, P. (eds) Modern Approaches to Discrete Curvature. Lecture Notes in Mathematics, vol 2184. Springer, Cham. https://doi.org/10.1007/978-3-319-58002-9_9

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