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.
Similar content being viewed by others
References
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)
Barvinok, A.: A Course in Convexity. American Mathematical Society, Providence, RI (2002)
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)
Chazal, F., Cohen-Steiner, D., Mérigot, Q.: Boundary measures for geometric inference. Found. Comput. Math. 10, 221–240 (2010)
Chazelle, B.: An optimal convex hull algorithm in any fixed dimension. Discrete Comput. Geom. 10, 377–409 (1993)
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)
Fremlin, D.H.: Skeletons and central sets. Proc. Lond. Math. Soc. 74, 701–720 (1997)
Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)
Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)
Hörrmann, J., Kousholt, A.: Reconstruction of convex bodies from moments. arXiv:1605.06362v1 (2016)
Hug, D., Schneider, R.: Local tensor valuations. Geom. Funct. Anal. 24, 1516–1564 (2014)
Hug, D., Last, G., Weil, W.: A local Steiner-type formula for general closed sets and applications. Math. Z. 246, 237–272 (2004)
Hug, D., Schneider, R., Schuster, R.: Integral geometry of tensor valuations. Adv. Appl. Math. 41, 482–509 (2008)
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)
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)
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)
Klette, R., Rosenfeld, A.: Digital Geometry. Elsevier, San Francisco (2004)
Kousholt, A., Kiderlen, M.: Reconstruction of convex bodies from surface tensors. Adv. Appl. Math. 76, 1–33 (2016)
Lindblad, J.: Surface area estimation of digitized 3D objects using weighted local configurations. Image Vis. Comput. 23, 111–122 (2005)
McMullen, P.: Isometry covariant valuations on convex bodies. Rend. Circ. Mat. Palermo 2(Suppl. 50), 259–271 (1997)
Mérigot, Q., Ovsjanikov, M., Guibas, L.: Voronoi-based curvature and feature estimation from point clouds. IEEE Trans. Vis. Comput. Graph. 17, 743–756 (2010)
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)
Mrkvička, T., Rataj, J.: On the estimation of intrinsic volume densities of stationary random closed sets. Stochastic Process. Appl. 118, 213–231 (2008)
Ohser, J., Mücklich, F.: Statistical Analysis of Microstructures. Wiley, Chichester (2000)
Ohser, J., Schladitz, K.: 3D Images of Materials Structures: Processing and Analysis. Wiley-VCH, Weinheim (2009)
Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, 2nd edn. Cambridge University Press, Cambridge (2014)
Schneider, R., Schuster, R.: Tensor valuations on convex bodies and integral geometry. II. Rend. Circ. Mat. Palermo 2(Suppl. 70), 295–314 (2002)
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)
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)
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)
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)
Svane, A.M.: On multigrid convergence of local algorithms for intrinsic volumes. J. Math. Imaging Vis. 49, 352–376 (2014)
Svane, A.M.: Estimation of Minkowski tensors from digital grey-scale images. Image Anal. Stereol. 34, 51–61 (2015)
Svane, A.M.: Local digital algorithms for estimating the integrated mean curvature of \(r\)-regular sets. Discrete Comput. Geom. 54, 316–338 (2015)
Zähle, M.: Integral and current representation of Federer’s curvature measures. Arch. Math. (Basel) 46, 557–567 (1986)
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)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor in Charge: Kenneth Clarkson
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-016-9851-x