Skip to main content
SpringerLink
Log in

Properties of Gauss Digitized Shapes and Digital Surface Integration

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

This paper presents new topological and geometric properties of Gauss digitizations of Euclidean shapes, most of them holding in arbitrary dimension d. We focus on r-regular shapes sampled by Gauss digitization at gridstep h. The digitized boundary is shown to be close to the Euclidean boundary in the Hausdorff sense, the minimum distance \(\frac{\sqrt{d}}{2}h\) being achieved by the projection map \(\xi \) induced by the Euclidean distance. Although it is known that Gauss digitized boundaries may not be manifold when \(d \ge 3\), we show that non-manifoldness may only occur in places where the normal vector is almost aligned with some digitization axis, and the limit angle decreases with h. We then have a closer look at the projection of the digitized boundary onto the continuous boundary by \(\xi \). We show that the size of its non-injective part tends to zero with h. This leads us to study the classical digital surface integration scheme, which allocates a measure to each surface element that is proportional to the cosine of the angle between an estimated normal vector and the trivial surface element normal vector. We show that digital integration is convergent whenever the normal estimator is multigrid convergent, and we explicit the convergence speed. Since convergent estimators are now available in the literature, digital integration provides a convergent measure for digitized objects.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Attali, D., Lieutier, A.: Reconstructing shapes with guarantees by unions of convex sets. In: Proceedings of the Twenty-Sixth Annual Symposium on Computational Geometry, pp. 344–353. ACM (2010)

  2. Chazal, F., Cohen-Steiner, D., Lieutier, A.: Normal cone approximation and offset shape isotopy. Comput. Geom. 42(6), 566–581 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chazal, F., Cohen-Steiner, D., Lieutier, A., Thibert, B.: Stability of curvature measures. In: Computer Graphics Forum, vol. 28, pp. 1485–1496. Wiley, New York (2009)

  4. Chazal, F., Cohen-Steiner, D., Mérigot, Q.: Boundary measures for geometric inference. Found. Comput. Math. 10(2), 221–240 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chazal, F., Lieutier, A.: Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees. Comput. Geom. 40(2), 156–170 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coeurjolly, D., Flin, F., Teytaud, O., Tougne, L.: Multigrid convergence and surface area estimation. In: Geometry, Morphology, and Computational Imaging, pp. 101–119. Springer, Berlin (2003)

  7. Coeurjolly, D., Lachaud, J.-O., Levallois, J.: Integral based curvature estimators in digital geometry. In: Discrete Geometry for Computer Imagery, Number 7749 in LNCS, pp. 215–227. Springer, Berlin (2013)

  8. Coeurjolly, D., Lachaud, J.-O., Levallois, J.: Multigrid convergent principal curvature estimators in digital geometry. Comput. Vis. Image Underst. 129, 27–41 (2014)

  9. Coeurjolly, D., Lachaud, J.-O., Roussillon, T.: Multigrid convergence of discrete geometric estimators. In: Brimkov, V.E., Barneva, R.P. (eds.) Digital Geometry Algorithms. Lecture Notes in Computational Vision and Biomechanics, vol. 2, pp. 395–424. Springer, Netherlands (2012)

    Google Scholar 

  10. Cuel, L., Lachaud, J.-O., Thibert, B.: Voronoi-based geometry estimator for 3d digital surfaces. In: Proceedings of the Discrete Geometry for Computer Imagery (DGCI’2014), Sienna, Italy. Lecture Notes in Computer Science, vol. 8668, pp. 134–149. Springer, Berlin (2014)

  11. de Vieilleville, F., Lachaud, J.-O., Feschet, F.: Maximal digital straight segments and convergence of discrete geometric estimators. J. Math. Image Vis. 27(2), 471–502 (2007)

    Google Scholar 

  12. Esbelin, H.-A., Malgouyres, R.: Convergence of binomial-based derivative estimation for c2-noisy discretized curves. In: Proceedings of the 15th DGCI. LNCS, vol. 5810, pp. 57–66 (2009)

  13. Esbelin, H.-A., Malgouyres, R., Cartade, C.: Convergence of binomial-based derivative estimation for \(c^2\) noisy discretized curves. Theor. Comput. Sci. 412(36), 4805–4813 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Federer, H.: Curvature measures. Trans. Am. Math. Soc 93(3), 418–491 (1959)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  16. Giraldo, A., Gross, A., Latecki, L.J.: Digitizations preserving shape. Pattern Recogn. 32(3), 365–376 (1999)

    Article  Google Scholar 

  17. Gross, A., Latecki, L.: Digitizations preserving topological and differential geometric properties. Comput. Vis. Image Underst. 62(3), 370–381 (1995)

    Article  Google Scholar 

  18. Huxley, M.N.: Exponential sums and lattice points. Proc. Lond. Math. Soc. 60, 471–502 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kenmochi, Y., Klette, R.: Surface area estimation for digitized regular solids. In: International Symposium on Optical Science and Technology, pp. 100–111. International Society for Optics and Photonics (2000)

  20. Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann Publishers Inc., San Francisco, CA (2004)

    Google Scholar 

  21. Klette, R., Sun, H.J.: Digital planar segment based polyhedrization for surface area estimation. In: Visual form 2001, pp. 356–366. Springer, Berlin (2001)

  22. Klette, R., Žunić, J.: Multigrid convergence of calculated features in image analysis. J. Math. Imaging Vis. 13(3), 173–191 (2000)

    Article  MATH  Google Scholar 

  23. Lachaud, J.-O.: Espaces non-euclidiens et analyse d’image : modèles déformables riemanniens et discrets, topologie et géométrie discrète. Habilitation à diriger des recherches, Université Bordeaux 1, Talence, France (2006)

  24. Lachaud, J.-O., Vialard, A., de Vieilleville, F.: Fast, accurate and convergent tangent estimation on digital contours. Image Vis. Comput. 25(10), 1572–1587 (2007)

    Article  Google Scholar 

  25. Latecki, L.J.: 3D well-composed pictures. Gr. Models Image Process. 59(3), 164–172 (1997)

    Article  Google Scholar 

  26. Latecki, L.J., Conrad, C., Gross, A.: Preserving topology by a digitization process. J. Math. Imaging Vis. 8(2), 131–159 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lenoir, A., Malgouyres, R., Revenu, M.: Fast computation of the normal vector field of the surface of a 3-d discrete object. In: Discrete Geometry for Computer Imagery, pp. 101–112. Springer, Berlin (1996)

  28. Lindblad, J.: Surface area estimation of digitized 3d objects using weighted local configurations. Image Vis. Comput. 23(2), 111–122 (2005)

    Article  MathSciNet  Google Scholar 

  29. Liu, Y.-S., Yi, J., Zhang, H., Zheng, G.-Q., Paul, J.-C.: Surface area estimation of digitized 3d objects using quasi-monte carlo methods. Pattern Recogn. 43(11), 3900–3909 (2010)

    Article  MATH  Google Scholar 

  30. Meine, H., Köthe, U., Stelldinger, P.: A topological sampling theorem for robust boundary reconstruction and image segmentation. Discrete Appl. Math. 157(3), 524–541 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mérigot, Q., Ovsjanikov, M., Guibas, L.: Voronoi-based curvature and feature estimation from point clouds. IEEE Trans. Vis. Comput. Gr. 17(6), 743–756 (2011)

    Article  Google Scholar 

  32. Morvan, J.-M.: Generalized Curvatures, vol. 2. Springer, Berlin (2008)

    MATH  Google Scholar 

  33. Niyogi, P., Smale, S., Weinberger, S.: Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39(1–3), 419–441 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pavlidis, T.: Algorithms for Graphics and Image Processing. Computer Science Press, Rockville, MD (1982)

    Book  Google Scholar 

  35. Provot, L., Gérard, Y.: Estimation of the derivatives of a digital function with a convergent bounded error. In: Discrete Geometry for Computer Imagery, pp. 284–295. Springer, Berlin (2011)

  36. Ronse, C., Tajine, M.: Discretization in hausdorff space. J. Math. Imaging Vis. 12(3), 219–242 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  37. Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)

    MATH  Google Scholar 

  38. Siqueira, M., Latecki, L.J., Tustison, N., Gallier, J., Gee, J.: Topological repairing of 3d digital images. J. Math. Imaging Vis. 30(3), 249–274 (2008)

    Article  MathSciNet  Google Scholar 

  39. Sloboda, F., Stoer, J.: On piecewise linear approximation of planar Jordan curves. J. Comput. Appl. Math. 55(3), 369–383 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  40. Stelldinger, P., Köthe, U.: Towards a general sampling theory for shape preservation. Image Vis. Comput. 23(2), 237–248 (2005)

    Article  Google Scholar 

  41. 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), 126–140 (2007)

    Article  Google Scholar 

  42. Tajine, M., Ronse, C.: Topological properties of hausdorff discretization, and comparison to other discretization schemes. Theor. Comput. Sci. 283(1), 243–268 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  43. Weyl, H.: On the volume of tubes. Am. J. Math. 61(2), 461–472 (1939)

    Article  MathSciNet  Google Scholar 

  44. Ziegel, J., Kiderlen, M.: Estimation of surface area and surface area measure of three-dimensional sets from digitizations. Image Vis. Comput. 28(1), 64–77 (2010)

    Article  Google Scholar 

Download references

Acknowledgments

This work was partially supported by the ANR grants DigitalSnow ANR-11- BS02-009, KIDICO ANR-2010-BLAN-0205 and TopData ANR-13-BS01-0008.

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques (LAMA), UMR 5127 CNRS, Université Savoie Mont Blanc, Chambéry, France

    Jacques-Olivier Lachaud

  2. Laboratoire Jean Kuntzman (LJK), UMR 5224 CNRS, Université Grenoble Alpes, Saint-Martin-d’Hères, France

    Boris Thibert

Authors
  1. Jacques-Olivier Lachaud
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Boris Thibert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jacques-Olivier Lachaud.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lachaud, JO., Thibert, B. Properties of Gauss Digitized Shapes and Digital Surface Integration. J Math Imaging Vis 54, 162–180 (2016). https://doi.org/10.1007/s10851-015-0595-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-015-0595-7

Keywords

Mathematics Subject Classification

Search

Navigation