Skip to main content
SpringerLink
Log in

Multigrid Convergence of Discrete Geometric Estimators

  • Chapter
Digital Geometry Algorithms

Part of the book series: Lecture Notes in Computational Vision and Biomechanics ((LNCVB,volume 2))

Abstract

The analysis of digital shapes requires tools to determine accurately their geometric characteristics. Their boundary is by essence discrete and is seen by continuous geometry as a jagged continuous curve, either straight or not derivable. Discrete geometric estimators are specific tools designed to determine geometric information on such curves. We present here global geometric estimators of area, length, moments, as well as local geometric estimators of tangent and curvature. We further study their multigrid convergence, a fundamental property which guarantees that the estimation tends toward the exact one as the sampling resolution gets finer and finer. Known theorems on multigrid convergence are summarized. A representative subsets of estimators have been implemented in a common framework (the library DGtal), and have been experimentally evaluated for several classes of shapes. Thus, the interested users have all the information for choosing the best adapted estimators to their applications, and readily dispose of an efficient implementation.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    DGtal uses a generic programming paradigm based on concepts and models of concepts. If a structure name starts from a capital “C”, we describe a concept.

References

  1. Amenta, N., Kil, Y.: Defining point-set surfaces. In: ACM SIGGRAPH, vol. 23, p. 270. ACM, New York (2004)

    Google Scholar 

  2. Asano, T., Kawamura, Y., Klette, R., Obokata, K.: Minimum-length polygons in approximation sausages. In: 4th International Workshop on Visual Form. Lecture Notes in Computer Science, vol. 2059, pp. 103–112. Springer, Berlin (2001)

    Google Scholar 

  3. Brlek, S., Labelle, G., Lacasse, A.: The discrete green theorem and some applications in discrete geometry. Theor. Comput. Sci. 346(2–3), 200–225 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bullard, J.W., Garboczi, E.J., Carter, W.C., Fullet, E.R.: Numerical methods for computing interfacial mean curvature. Comput. Mater. Sci. 4, 103–116 (1995)

    Article  Google Scholar 

  5. Cazals, F., Pouget, M.: Estimating differential quantities using polynomial fitting of osculating jets. Comput. Aided Geom. Des. 22, 121–146 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chazal, F., Cohen-Steiner, D., Lieutier, A., Thibert, B.: Stability of Curvature Measures (2008)

    Google Scholar 

  7. Chen, K.: Adaptive smoothing via contextual and local discontinuities. IEEE Trans. Pattern Anal. Mach. Intell. 27(10), 1552–1566 (2005)

    Article  Google Scholar 

  8. Coeurjolly, D.: Algorithmique et géométrie pour la caractérisation des courbes et des surfaces. Ph.D. thesis, Université Lyon 2 (2002)

    Google Scholar 

  9. 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”. Lecture Notes in Computer Science, vol. 2616, pp. 101–119. Springer, Berlin (2003)

    Google Scholar 

  10. Coeurjolly, D., Gérard, Y., Reveillès, J.P., Tougne, L.: An elementary algorithm for digital arc segmentation. Discrete Appl. Math. 139(1–3), 31–50 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Coeurjolly, D., Klette, R.: A comparative evaluation of length estimators of digital curves. IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 252–258 (2004)

    Article  Google Scholar 

  12. Coeurjolly, D., Miguet, S., Tougne, L.: Discrete curvature based on osculating circle estimation. In: 4th International Workshop on Visual Form. Lecture Notes in Computer Science, vol. 2059, pp. 303–312 (2001)

    Google Scholar 

  13. Coeurjolly, D., Svensson, S.: Estimation of curvature along curves with application to fibres in 3d images of paper. In: 13th Scandinavian Conference on Image Analysis. Lecture Notes in Computer Science, vol. 2749, pp. 247–254 (2003)

    Google Scholar 

  14. Debled-Rennesson, I., Feschet, F., Rouyer-Degli, J.: Optimal blurred segments decomposition of noisy shapes in linear times. Comput. Graph. 30, 30–36 (2006)

    Article  Google Scholar 

  15. de Vieilleville, F., Lachaud, J.O.: Experimental comparison of continuous and discrete tangent estimators along digital curves. In: 12th International Workshop on Combinatorial Image Analysis. Lecture Notes in Computer Science, vol. 4958, pp. 26–37. Springer, Berlin (2008)

    Chapter  Google Scholar 

  16. de Vieilleville, F., Lachaud, J.O.: Comparison and improvement of tangent estimators on digital curves. Pattern Recognit. 42(8), 1693–1707 (2009)

    Article  MATH  Google Scholar 

  17. de Vieilleville, F., Lachaud, J.O., Feschet, F.: Convex digital polygons, maximal digital straight segments and convergence of discrete geometric estimators. J. Math. Imaging Vis. 27(2), 139–156 (2007)

    Article  Google Scholar 

  18. DGtal: digital geometry tools and algorithms library. http://liris.cnrs.fr/dgtal

  19. Dorst, L., Smeulders, A.W.M.: Length estimators for digitized contours. Comput. Vis. Graph. Image Process. 40(3), 311–333 (1987)

    Article  Google Scholar 

  20. Elder, J., Zucker, S.W.: Local scale control for edge detection and blur estimation. IEEE Trans. Pattern Anal. Mach. Intell. 20(7), 669–716 (1998)

    Article  Google Scholar 

  21. Esbelin, H.A., Malgouyres, R.: Convergence of binomial-based derivative estimation for c2-noisy discretized curves. In: 15th Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 5810, pp. 57–66 (2009)

    Chapter  Google Scholar 

  22. Feschet, F., Tougne, L.: Optimal time computation of the tangent of a discrete curve: application to the curvature. In: Proc. 8th Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 1568, pp. 31–40 (1999)

    Chapter  Google Scholar 

  23. Fiorio, C., Mercat, C., Rieux, F.: Curvature estimation for discrete curves based on auto-adaptive masks of convolution. In: Computational Modeling of Objects Presented in Images. Lecture Notes in Computer Science, vol. 6026, pp. 47–59 (2010)

    Chapter  Google Scholar 

  24. Fleischmann, O., Wietzke, L., Sommer, G.: A novel curvature estimator for digital curves and images. In: 32th Annual Symposium of the German Association for Pattern Recognition. Lecture Notes in Computer Science, vol. 6376, pp. 442–451 (2010)

    Google Scholar 

  25. Gérard, Y., Provot, L., Feschet, F.: Introduction to digital level layers. In: 16th IAPR International Conference Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 6607, pp. 83–94. Springer, Berlin (2011)

    Chapter  Google Scholar 

  26. Goshtasby, A., Satter, M.: An adaptive window mechanism for image smoothing. Comput. Vis. Image Underst. 111, 155–169 (2008)

    Article  Google Scholar 

  27. Hermann, S., Klette, R.: A comparative study on 2d curvature estimators. In: 17th International Conference on Computer Theory and Applications, pp. 584–589 (2007)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. ImaGene: generic digital image library. https://gforge.liris.cnrs.fr/projects/imagene

  30. Kerautret, B., Lachaud, J.: Multiscale analysis of discrete contours for unsupervised noise detection. In: 13th International Workshop on Combinatorial Image Analysis. Lecture Notes in Computer Science, vol. 5852, pp. 187–200. Springer, Mexico (2009)

    Chapter  Google Scholar 

  31. Kerautret, B., Lachaud, J.: Meaningful scales detection along digital contours for unsupervised local noise estimation. IEEE Trans. Pattern Anal. Mach. Intell. (submitted)

    Google Scholar 

  32. Kerautret, B., Lachaud, J.O.: Curvature estimation along noisy digital contours by approximate global optimization. Pattern Recognit. 42(10), 2265–2278 (2009)

    Article  MATH  Google Scholar 

  33. Kerautret, B., Lachaud, J.O.: Noise level and meaningful scale detection online demonstration. http://kerrecherche.iutsd.uhp-nancy.fr/MeaningfulBoxes (2010)

  34. Kervrann, C.: An adaptive window approach for image smoothing and structures preserving. In: 8th European Conference on Computer Vision. Lecture Notes in Computer Science, vol. 3023, pp. 132–144. Springer, Berlin (2004)

    Google Scholar 

  35. Klette, R., Rosenfeld, A.: Digital Geometry—Geometric Methods for Digital Picture Analysis (2004)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  37. Kovalevsky, V., Fuchs, S.: Theoretical and experimental analysis of the accuracy of perimeter estimates. In: Proc. Robust Computer Vision, pp. 218–242 (1992)

    Google Scholar 

  38. Kovalevsky, V.A.: New definition and fast recognition of digital straight segments and arcs. In: 10th International Conference on Pattern Analysis and Pattern Recognition, pp. 31–34 (1990)

    Google Scholar 

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

    Google Scholar 

  40. Lachaud, J.O., Vialard, A.: Geometric measures on arbitrary dimensional digital surfaces. In: 11th International Conference Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 2886, pp. 434–443. Springer, Berlin (2003)

    Chapter  Google Scholar 

  41. Lachaud, J.O., Vialard, A., de Vieilleville, F.: Analysis and comparative evaluation of discrete tangent estimators. In: 12th International Conference on Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 3429, pp. 140–251. Springer, Berlin (2005)

    Google Scholar 

  42. 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 

  43. 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 

  44. Lenoir, A.: Fast estimation of mean curvature on the surface of a 3d discrete object. In: 7th International Workshop on Discrete Geometry for Computer Imagery, pp. 175–186. Springer, Berlin (1997)

    Chapter  Google Scholar 

  45. Lenoir, A., Malgouyres, R., Revenu, M.: Fast computation of the normal vector field of the surface of a 3d discrete object. In: 6th International Workshop on Discrete Geometry for Computer Imagery, vol. 1176, pp. 101–109 (1996)

    Chapter  Google Scholar 

  46. Lewiner, T., Gomes, J.D., Jr., Lopes, H., Craizer, M.: Curvature and torsion estimators based on parametric curve fitting. Comput. Graph. 29, 641–655 (2005)

    Article  Google Scholar 

  47. Lien, S.L.C.: Combining computation with geometry. Ph.D. thesis, California Institute of Technology (1984)

    Google Scholar 

  48. 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 

  49. Liu, H., Latecki, L., Liu, W.: A unified curvature definition for regular, polygonal, and digital planar curves. Int. J. Comput. Vis. 80(1), 104–124 (2008)

    Article  Google Scholar 

  50. 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 Recognit. 43(11), 3900–3909 (2010)

    Article  MATH  Google Scholar 

  51. Malgouyres, R., Brunet, F., Fourey, S.: Binomial convolutions and derivatives estimation from noisy discretizations. In: 14th Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 4992, pp. 370–379 (2008)

    Chapter  Google Scholar 

  52. Marji, M.: On the detection of dominant points on digital planar curves. Ph.D. thesis, Wayne State University, Detroit (2003)

    Google Scholar 

  53. Matas, J., Shao, Z., Kittler, J.: Estimation of curvature and tangent direction by median filtered differencing. In: 8th International Conference on Image Analysis and Processing. Lecture Notes in Computer Science, vol. 974, pp. 83–88 (1995)

    Google Scholar 

  54. Merigot, Q., Ovsjanikov, M., Guibas, L.: Robust Voronoi-based curvature and feature estimation. In: SIAM/ACM Joint Conference on Geometric and Physical Modeling (2009)

    Google Scholar 

  55. Nguyen, T., Debled-Rennesson, I.: Curvature estimation in noisy curves. In: 12th International Conference on Computer Analysis of Images and Patterns. Lecture Notes in Computer Science, vol. 4673, pp. 474–481 (2007)

    Chapter  Google Scholar 

  56. Novotni, M., Klein, R.: Shape retrieval using 3D Zernike descriptors. Comput. Aided Des. 36(11), 1047–1062 (2004)

    Article  Google Scholar 

  57. Pottmann, H., Wallner, J., Huang, Q., Yang, Y.: Integral invariants for robust geometry processing. Comput. Aided Geom. Des. 26(1), 37–60 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  58. Pottmann, H., Wallner, J., Yang, Y., Lai, Y., Hu, S.: Principal curvatures from the integral invariant viewpoint. Comput. Aided Geom. Des. 24(8–9), 428–442 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  59. Proffitt, D., Rosen, D.: Metrication errors and coding efficiency of chain-encoding schemes for the representation of lines and edges. Comput. Graph. Image Process. 10(4), 318–332 (1979)

    Article  Google Scholar 

  60. Provençal, X., Lachaud, J.O.: Two linear-time algorithms for computing the minimum length polygon of a digital contour. In: 15th International Conference on Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 5810, pp. 104–117 (2009)

    Chapter  Google Scholar 

  61. Provot, L., Gérard, Y.: Estimation of the derivatives of a digital function with a convergent bounded error. In: 16th IAPR International Conference in Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 6607, pp. 284–295. Springer, Berlin (2011)

    Chapter  Google Scholar 

  62. Roussillon, T., Lachaud, J.O.: Accurate curvature estimation along digital contours with maximal digital circular arcs. In: 14th International Workshop in Combinatorial Image Analysis. Lecture Notes in Computer Science, vol. 6636, pp. 43–55. Springer, Berlin (2011)

    Chapter  Google Scholar 

  63. Roussillon, T., Sivignon, I.: Faithful polygonal representation of the convex and concave parts of a digital curve. Pattern Recognit. 44(10–11), 2693–2700 (2011)

    Article  MATH  Google Scholar 

  64. Roussillon, T., Sivignon, I., Tougne, L.: On three constrained versions of the digital circular arc recognition problem. In: 15th IAPR International Conference on Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 5810, pp. 34–45 (2009)

    Chapter  Google Scholar 

  65. Sloboda, F., Zat̆ko, B., Stoer, J.: On approximation of planar one-dimensional continua. In: Advances in Digital and Computational Geometry, pp. 113–160 (1998)

    Google Scholar 

  66. Tajine, M., Daurat, A.: On local definitions of length of digital curves. In: 11th International Conference on Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 2886, pp. 114–123. Springer, Berlin (2003)

    Chapter  Google Scholar 

  67. Teague, M.R.: Image analysis via the general theory of moments. J. Opt. Soc. Am. 70(8), 920–930 (1980)

    Article  MathSciNet  Google Scholar 

  68. Vialard, A.: Geometrical parameters extraction from discrete paths. In: 7th Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 1176, pp. 24–35 (1996)

    Chapter  Google Scholar 

  69. Windreich, G., Kiryati, N., Lohmann, G.: Voxel-based surface area estimation: from theory to practice. Pattern Recognit. 36(11), 2531–2541 (2003)

    Article  MATH  Google Scholar 

  70. Worring, M., Smeulders, A.W.M.: Digital curvature estimation. CVGIP, Image Underst. 58(3), 366–382 (1993)

    Article  Google Scholar 

Download references

Acknowledgements

This work was partially funded by project KIDICO (ANR-2010-BLAN-0205) of the French Research Agency (ANR).

Author information

Authors and Affiliations

  1. LIRIS, UMR CNRS 5205, Université de Lyon, 69676, Villeurbanne, France

    David Coeurjolly & Tristan Roussillon

  2. LAMA, UMR CNRS 5127, University of Savoie, 73376, Le Bourget du Lac, France

    Jacques-Olivier Lachaud

Authors
  1. David Coeurjolly
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jacques-Olivier Lachaud
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tristan Roussillon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Coeurjolly .

Editor information

Editors and Affiliations

  1. Buffalo State College, Department of Mathematics, State University of New York, Elmwood Avenue 1300, Buffalo, 14222, New York, USA

    Valentin E. Brimkov

  2. Department of Computer Science, State University of New York at Fredonia, Central Ave. 280, Fredonia, 14063, New York, USA

    Reneta P. Barneva

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Coeurjolly, D., Lachaud, JO., Roussillon, T. (2012). Multigrid Convergence of Discrete Geometric Estimators. In: Brimkov, V., Barneva, R. (eds) Digital Geometry Algorithms. Lecture Notes in Computational Vision and Biomechanics, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4174-4_13

Download citation

  • DOI: https://doi.org/10.1007/978-94-007-4174-4_13

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-4173-7

  • Online ISBN: 978-94-007-4174-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation