Advertisement

Journal of Mathematical Imaging and Vision

, Volume 49, Issue 1, pp 148–172 | Cite as

On Multigrid Convergence of Local Algorithms for Intrinsic Volumes

  • Anne Marie Svane
Article

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.

Keywords

Image analysis Local algorithm Multigrid convergence Intrinsic volumes Binary morphology 

Notes

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.

References

  1. 1.
    Aomoto, K.: Analytic structure of Schläfli function. Nagoya Math. J. 68, 1–16 (1977) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Coeurjolly, D., Flin, F., Teytaud, O., Tougne, L.: Multigrid convergence and surface area estimation. In: Theoretical Foundations of Computer Vision Geometry, Morphology, and Computational Imaging. LNCS, vol. 2616, pp. 101–119. Springer, Berlin (2003) Google Scholar
  3. 3.
    Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Freeman, H.: Boundary encoding and processing. In: Lipkin, B.S., Rosenfeld, A. (eds.) Picture Processing and Psychopictorics, pp. 241–266. Academic Press, New York (1970) Google Scholar
  5. 5.
    Kampf, J.: A limitation of the estimation of intrinsic volumes via pixel configuration counts. WiMa Report 144 (2012) Google Scholar
  6. 6.
    Kenmochi, Y., Klette, R.: Surface area estimation for digitized regular solids. In: Latecki, L.J., Mount, D.M., Wu, A.Y. (eds.) Proceedings of SPIE, vol. 4117, pp. 100–111. Vision Geometry IX, San Diego (2000) Google Scholar
  7. 7.
    Kiderlen, M., Rataj, J.: On infinitesimal increase of volumes of morphological transforms. Mathematika 53(1), 103–127 (2007) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Klette, R., Rosenfeld, A.: Digital Geometry. Elsevier, San Fransisco (2004) zbMATHGoogle Scholar
  9. 9.
    Klette, R., Sun, H.J.: Digital planar segment based polyhedrization for surface area estimation. In: Arcelli, C., Cordella, L.P., Sanniti di Baja, G. (eds.) Visual Form 2001. LNCS, vol. 2059, pp. 356–366. Springer, Capri (2001) CrossRefGoogle Scholar
  10. 10.
    Lee, C.-N., Poston, T., Rosenfeld, A.: Winding and Euler numbers for 2D and 3D digital images. Comput. Vis. Graph. Image Process. 53(6), 522–537 (1991) zbMATHGoogle Scholar
  11. 11.
    Lindblad, J.: Surface area estimation of digitized 3D objects using weighted local configurations. Image Vis. Comput. 23, 111–122 (2005) CrossRefGoogle Scholar
  12. 12.
    Mecke, K.R.: Morphological characterization of patterns in reaction-diffusion systems. Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53, 4794–4800 (1996) Google Scholar
  13. 13.
    Ohser, J., Mücklich, F.: Statistical Analysis of Microstructures. Wiley, Chichester (2000) zbMATHGoogle Scholar
  14. 14.
    Ohser, J., Sandau, K., Kampf, J., Vecchio, I., Moghiseh, A.: Improved estimation of fiber length from 3-dimensional images. Image Anal. Stereol. 32, 45–55 (2013) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Pavlidis, T.: Algorithms for Graphics and Image Processing. Comput. Sci. Press, New York (1982) CrossRefGoogle Scholar
  16. 16.
    Schläfli, L.: On the multiple integral ∫n dxdydz whose limits are p 1=a 1 x+b 1 y+⋯+h 1 z>0, p 2>0,…,P n>0, and x 2+y 2+⋯+z 2<1. Pure Appl. Math. Q. 2, 269–301 (1858) Google Scholar
  17. 17.
    Schneider, R.: Convex Bodies: the Brunn–Minkowski Theory. Cambridge University Press, Cambridge (1993) CrossRefzbMATHGoogle Scholar
  18. 18.
    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) CrossRefGoogle Scholar
  19. 19.
    Schröder-Turk, G.E., Mickel, W., Kapfer, S.C., Schaller, F.M., Breidenbach, B., Hug, D., Mecke, K.: Minkowski tensors of anisotropic spatial structure (2010). arXiv:1009.2340
  20. 20.
    Serra, J.: Image Analysis and Mathematical Morphology vol. 1. Academic Press, San Diego (1984) Google Scholar
  21. 21.
    Stelldinger, P., Latecki, L.J., Siqueira, M.: Topological equivalence between a 3D object and the reconstruction of its digital image. IEEE Trans. Pattern Anal. Mach. Intell. 29(1), 1–15 (2007) CrossRefGoogle Scholar
  22. 22.
    Svane, A.M.: Local digital algorithms for estimating the integrated mean curvature of r-regular sets. CSGB Research Report no. 8, version 2 (2012) Google Scholar
  23. 23.
    Tajine, M., Daurat, A.: On local definitions of length of digital curves. In: Sanniti di Baja, G., Svensson, S., Nyström, I. (eds.) Discrete Geometry for Computer Imagery. LNCS, vol. 2886, pp. 114–123. Springer, Berlin (2003) CrossRefGoogle Scholar
  24. 24.
    Ziegel, J., Kiderlen, M.: Estimation of surface area and surface area measure of three-dimensional sets from digitizations. Image Vis. Comput. 28, 64–77 (2010) CrossRefGoogle Scholar
  25. 25.
    Ziegler, G.: Lectures on Polytopes. Springer, New York (1995) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsAarhus UniversityAarhus CDenmark

Personalised recommendations