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Voronoi-Based Estimation of Minkowski Tensors from Finite Point Samples

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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.

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References

  1. Arns, C.A., Knackstedt, M.A., Mecke, K.R.: Characterising the morphology of disordered materials. In: Mecke, K.R., Stoyan, D. (eds.) Morphology of Condensed Matter. Lecture Notes in Physics, vol. 600, pp. 37–74. Springer, Berlin (2002)

    Chapter  Google Scholar 

  2. Barvinok, A.: A Course in Convexity. American Mathematical Society, Providence, RI (2002)

  3. Beisbart, C., Barbosa, M.S., Wagner, H., da Costa, L.F.: Extended morphometric analysis of neuronal cells with Minkowski valuations. Eur. Phys. J. B 52, 531–546 (2006)

    Article  Google Scholar 

  4. Chazal, F., Cohen-Steiner, D., Mérigot, Q.: Boundary measures for geometric inference. Found. Comput. Math. 10, 221–240 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chazelle, B.: An optimal convex hull algorithm in any fixed dimension. Discrete Comput. Geom. 10, 377–409 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Clarenz, U., Rumpf, M., Telea, A.: Robust feature detection and local classification for surfaces based on moment analysis. IEEE Trans. Vis. Comput. Graph. 10, 516–524 (2004)

    Article  Google Scholar 

  7. Fremlin, D.H.: Skeletons and central sets. Proc. Lond. Math. Soc. 74, 701–720 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  9. Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)

    MATH  Google Scholar 

  10. Hörrmann, J., Kousholt, A.: Reconstruction of convex bodies from moments. arXiv:1605.06362v1 (2016)

  11. Hug, D., Schneider, R.: Local tensor valuations. Geom. Funct. Anal. 24, 1516–1564 (2014)

    Article  MathSciNet  Google Scholar 

  12. Hug, D., Last, G., Weil, W.: A local Steiner-type formula for general closed sets and applications. Math. Z. 246, 237–272 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hug, D., Schneider, R., Schuster, R.: Integral geometry of tensor valuations. Adv. Appl. Math. 41, 482–509 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kapfer, S.C., Mickel, W., Schaller, F.M., Spanner, M., Goll, C., Nogawa, T., Ito, N., Mecke, K., Schröder-Turk, G.E.: Local anisotropy of fluids using Minkowski tensors. J. Stat. Mech. Theory Exp. 2010, P11010 (2010)

    Article  Google Scholar 

  15. Kiderlen, M., Vedel Jensen, E.B.V. (eds.): Tensor Valuations and Their Applications in Stochastic Geometry and Imaging. Lecture Notes in Mathematics, vol. 2177. Springer, Berlin (2017)

  16. Klenk, S., Schmidt, V., Spodarev, E.: A new algorithmic approach to the computation of Minkowski functionals of polyconvex sets. Comput. Geom. 34, 127–148 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Klette, R., Rosenfeld, A.: Digital Geometry. Elsevier, San Francisco (2004)

    MATH  Google Scholar 

  18. Kousholt, A., Kiderlen, M.: Reconstruction of convex bodies from surface tensors. Adv. Appl. Math. 76, 1–33 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lindblad, J.: Surface area estimation of digitized 3D objects using weighted local configurations. Image Vis. Comput. 23, 111–122 (2005)

    Article  Google Scholar 

  20. McMullen, P.: Isometry covariant valuations on convex bodies. Rend. Circ. Mat. Palermo 2(Suppl. 50), 259–271 (1997)

    MathSciNet  MATH  Google Scholar 

  21. Mérigot, Q., Ovsjanikov, M., Guibas, L.: Voronoi-based curvature and feature estimation from point clouds. IEEE Trans. Vis. Comput. Graph. 17, 743–756 (2010)

    Article  Google Scholar 

  22. Miles, R.E., Serra, J. (eds.): Geometrical Probability and Biological Structures: Buffons 200th Anniversary (Proceedings Paris, 1977). Lecture Notes in Biomath, vol. 23, Springer, Berlin (1978)

  23. Mrkvička, T., Rataj, J.: On the estimation of intrinsic volume densities of stationary random closed sets. Stochastic Process. Appl. 118, 213–231 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ohser, J., Mücklich, F.: Statistical Analysis of Microstructures. Wiley, Chichester (2000)

    MATH  Google Scholar 

  25. Ohser, J., Schladitz, K.: 3D Images of Materials Structures: Processing and Analysis. Wiley-VCH, Weinheim (2009)

    Book  MATH  Google Scholar 

  26. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, 2nd edn. Cambridge University Press, Cambridge (2014)

    MATH  Google Scholar 

  27. Schneider, R., Schuster, R.: Tensor valuations on convex bodies and integral geometry. II. Rend. Circ. Mat. Palermo 2(Suppl. 70), 295–314 (2002)

    MathSciNet  MATH  Google Scholar 

  28. Schröder-Turk, G.E., Kapfer, S.C., Breidenbach, B., Beisbart, C., Mecke, K.: Tensorial Minkowski functionals and anisotropy measures for planar patterns. J. Microsc. 238, 57–74 (2008)

    Article  MathSciNet  Google Scholar 

  29. Schröder-Turk, G.E., Mickel, W., Schröter, M., Delaney, G.W., Saadatfar, M., Senden, T.J., Mecke, K., Aste, T.: Disordered spherical bead packs are anisotropic. Europhys. Lett. 90, 34001 (2010)

    Article  Google Scholar 

  30. Schröder-Turk, G.E., Mickel, W., Kapfer, S.C., Klatt, M.A., Schaller, F.M., Hoffmann, M.J., Kleppmann, N., Armstrong, P., Inayat, A., Hug, D., Reichelsdorfer, M., Peukert, W., Schwieger, W., Mecke, K.: Minkowski tensor shape analysis of cellular, granular and porous structures. Adv. Mater. 23, 2535–2553 (2011)

    Article  Google Scholar 

  31. Schröder-Turk, G.E., Mickel, W., Kapfer, S.C., Schaller, F.M., Breidenbach, B., Hug, D., Mecke, K.: Minkowski tensors of anisotropic spatial structure. New J. Phys. 15, 083028 (2013)

    Article  MathSciNet  Google Scholar 

  32. Svane, A.M.: On multigrid convergence of local algorithms for intrinsic volumes. J. Math. Imaging Vis. 49, 352–376 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Svane, A.M.: Estimation of Minkowski tensors from digital grey-scale images. Image Anal. Stereol. 34, 51–61 (2015)

    MathSciNet  MATH  Google Scholar 

  34. Svane, A.M.: Local digital algorithms for estimating the integrated mean curvature of \(r\)-regular sets. Discrete Comput. Geom. 54, 316–338 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zähle, M.: Integral and current representation of Federer’s curvature measures. Arch. Math. (Basel) 46, 557–567 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ziegel, J.F., Nyengaard, J.R., Jensen, E.B.V.: Estimating particle shape and orientation using volume tensors. Scand. J. Stat. 42, 813–831 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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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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Authors and Affiliations

  1. Department of Mathematics, Karlsruhe Institute of Technology, 76128, Karlsruhe, Germany

    Daniel Hug

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

    Markus Kiderlen & Anne Marie Svane

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  1. Daniel Hug
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  2. Markus Kiderlen
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  3. Anne Marie Svane
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Correspondence to Anne Marie Svane.

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Hug, D., Kiderlen, M. & Svane, A.M. Voronoi-Based Estimation of Minkowski Tensors from Finite Point Samples. Discrete Comput Geom 57, 545–570 (2017). https://doi.org/10.1007/s00454-016-9851-x

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  • DOI: https://doi.org/10.1007/s00454-016-9851-x

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