Journal of Mathematical Imaging and Vision

, Volume 61, Issue 3, pp 359–379 | Cite as

Laplace–Beltrami Operator on Digital Surfaces

  • Thomas CaissardEmail author
  • David Coeurjolly
  • Jacques-Olivier Lachaud
  • Tristan Roussillon


This article presents a novel discretization of the Laplace–Beltrami operator on digital surfaces. We adapt an existing convolution technique proposed by Belkin et al. (in: Teillaud (ed) Proceedings of the 24th ACM symposium on computational geometry, College Park, MD, USA, pp 278–287, 2008, for triangular meshes to topological border of subsets of \(\mathbb {Z}^n\). The core of the method relies on first-order estimation of measures associated with our discrete elements (such as length, area etc.). We show strong consistency (i.e., pointwise convergence) of the operator and compare it against various other discretizations.


Laplace–Beltrami operator Digital surface Discrete geometry Differential geometry 


Supplementary material


  1. 1.
    Alexa, M., Wardetzky, M.: Discrete Laplacians on general polygonal meshes. In: ACM SIGGRAPH 2011 Papers, SIGGRAPH’11, pp. 102:1–102:10. ACM, New York, NY, USA (2011).
  2. 2.
    Belkin, M., Niyogi, P.: Towards a theoretical foundation for Laplacian-based manifold methods. J. Comput. Syst. Sci. 74(8), 1289–1308 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Belkin, M., Sun, J., Wang, Y.: Constructing Laplace Operator from Point Clouds in \({\mathbb{R}}^d\), pp. 1031–1040.
  4. 4.
    Belkin, M., Sun, J., Wang, Y.: Discrete Laplace operator on meshed surfaces. In: Teillaud, M. (ed.) Proceedings of the 24th ACM Symposium on Computational Geometry, College Park, MD, USA, June 9–11, 2008, pp. 278–287. ACM (2008).
  5. 5.
    Caissard, T., Coeurjolly, D., Lachaud, J.O., Roussillon, T.: Heat kernel Laplace–Beltrami operator on digital surfaces. Working paper or preprint (2016)Google Scholar
  6. 6.
    Carl, W.: A Laplace operator on semi-discrete surfaces. Found. Comput. Math. 16(5), 1115–1150 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cartade, C., Mercat, C., Malgouyres, R., Samir, C.: Mesh parameterization with generalized discrete conformal maps. J. Math. Imaging Vis. 46(1), 1–11 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cazals, F., Pouget, M.: Estimating differential quantities using polynomial fitting of osculating jets. Comput. Aided Geom. Des. 22(2), 121–146 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Coeurjolly, D., Lachaud, J., Levallois, J.: Integral based curvature estimators in digital geometry. In: González-Díaz, R., Jiménez, M.J., Medrano, B. (eds.) Discrete Geometry for Computer Imagery—17th IAPR International Conference, DGCI 2013, Seville, Spain, March 20–22, 2013. Proceedings, Lecture Notes in Computer Science, vol. 7749, pp. 215–227. Springer (2013).
  10. 10.
    Coeurjolly, D., Lachaud, J., Levallois, J.: Multigrid convergent principal curvature estimators in digital geometry. Comput. Vis. Image Underst. 129, 27–41 (2014). CrossRefzbMATHGoogle Scholar
  11. 11.
    Coeurjolly, D., Lachaud, J.O., Roussillon, T.: Multigrid Convergence of Discrete Geometric Estimators, pp. 395–424. Springer, Dordrecht (2012).
  12. 12.
    Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the lambertw function. Adv. Comput. Math. 5(1), 329–359 (1996). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Crane, K., Weischedel, C., Wardetzky, M.: Geodesics in heat: a new approach to computing distance based on heat flow. ACM TOG 32(5), 152 (2013)CrossRefGoogle Scholar
  14. 14.
    Cuel, L., Lachaud, J.O., Thibert, B.: Voronoi-based geometry estimator for 3d digital surfaces. In: Proceedings of Discrete Geometry for Computer Imagery (DGCI’2014), LNCS, vol. 8668, pp. 134–149 (2014)Google Scholar
  15. 15.
    DGtal: Digital geometry tools and algorithms library.
  16. 16.
    Desbrun, M., Hirani, A.N., Leok, M., Marsden, J.E.: Discrete exterior calculus. arXiv preprint arXiv: math/0508341 (2005)
  17. 17.
    Desbrun, M., Meyer, M., Schröder, P., Barr, A.H.: Implicit fairing of irregular meshes using diffusion and curvature flow. In: Waggenspack, W.N. (ed.) Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 1999, Los Angeles, CA, USA, August 8–13, 1999, pp. 317–324. ACM (1999).
  18. 18.
    Dey, T.K., Ranjan, P., Wang, Y.: Convergence, Stability, and Discrete Approximation of Laplace Spectra, pp. 650–663.
  19. 19.
    Dziuk, G.: Finite Elements for the Beltrami Operator on Arbitrary Surfaces, pp. 142–155. Springer, Berlin (1988).
  20. 20.
    de Vieilleville, F., Lachaud, J., Feschet, F.: Convex digital polygons, maximal digital straight segments and convergence of discrete geometric estimators. J. Math. Imaging Vis. 27(2), 139–156 (2007). MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ellis, T., Proffitt, D., Rosen, D., Rutkowski, W.: Measurement of the lengths of digitized curved lines. Comput. Graph. Image Process. 10(4), 333–347 (1979). CrossRefGoogle Scholar
  22. 22.
    Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93(3), 418–491 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Flin, F., Brzoska, J.B., Lesaffre, B., Coléou, C., Lamboley, P., Coeurjolly, D., Teytaud, O., Vignoles, G., Delesse, J.F.: An adaptive filtering method to evaluate normal vectors and surface areas of 3d objects. Application to snow images from X-ray tomography. IEEE Trans. Image Process. 14(5), 585–596 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Fujiwara, K.: Eigenvalues of Laplacians on a closed Riemannian manifold and its nets. Proc. Am. Math. Soc. 123(8), 2585–2594 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Grady, L.J., Polimeni, J.: Discrete Calculus: Applied Analysis on Graphs for Computational Science. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  26. 26.
    Harrison, J.: Stokes’ theorem for nonsmooth chains. Bull. Am. Math. Soc. 29(2), 235–242 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Harrison, J.: Flux across nonsmooth boundaries and fractal Gauss/Green/Stokes’ theorems. J. Phys. A Math. Gen. 32(28), 5317 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hein, M., Audibert, J.Y., von Luxburg, U.: From Graphs to Manifolds—Weak and Strong Pointwise Consistency of Graph Laplacians, pp. 470–485. Springer, Berlin (2005).
  29. 29.
    Herman, G.: Geometry of Digital Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2012)Google Scholar
  30. 30.
    Hildebrand, T., Laib, A., Müller, R., Dequeker, J., Rüegsegger, P.: Direct three-dimensional morphometric analysis of human cancellous bone: microstructural data from spine, femur, iliac crest, and calcaneus. J. Bone Miner. Res. 14(7), 1167–74 (1999)CrossRefGoogle Scholar
  31. 31.
    Hildebrandt, K., Polthier, K.: Generalized shape operators on polyhedral surfaces. Comput. Aided Geom. Des. 28(5), 321–343 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hildebrandt, K., Polthier, K.: On approximation of the Laplace–Beltrami operator and the Willmore energy of surfaces. Comput. Graph. Forum 30(5), 1513–1520 (2011). CrossRefGoogle Scholar
  33. 33.
    Hildebrandt, K., Polthier, K., Wardetzky, M.: On the convergence of metric and geometric properties of polyhedral surfaces. Geom. Dedic. 123(1), 89–112 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Hirani, A.N.: Discrete exterior calculus. Ph.D. thesis, California Institute of Technology (2003)Google Scholar
  35. 35.
    Hunter, J., Nachtergaele, B.: Applied Analysis. World Scientific, Singapore (2001)CrossRefzbMATHGoogle Scholar
  36. 36.
    Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. The Morgan Kaufmann Series in Computer Graphics and Geometric Modeling. Elsevier, Amsterdam (2004)zbMATHGoogle Scholar
  37. 37.
    Lachaud, J., Vialard, A., de Vieilleville, F.: Fast, accurate and convergent tangent estimation on digital contours. Image Vis. Comput. 25(10), 1572–1587 (2007). CrossRefGoogle Scholar
  38. 38.
    Lachaud, J.O.: Non-Euclidean spaces and image analysis : Riemannian and discrete deformable models, discrete topology and geometry. Habilitation à diriger des recherches, Université Sciences et Technologies - Bordeaux I (2006)Google Scholar
  39. 39.
    Lachaud, J.O., Thibert, B.: Properties of Gauss digitized shapes and digital surface integration. J. Math. Imaging Vis. 54(2), 162–180 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Lenoir, A.: Des outils pour les surfaces discrètes : estimation d’invariants géométriques, préservation de la topologie, tracé de géodésiques et visualisation. Ph.D. thesis, Université de Caen (1999)Google Scholar
  41. 41.
    Lenoir, A., Malgouyres, R., Revenu, M.: Fast computation of the normal vector field of the surface of a 3-D discrete object, pp. 101–112. Springer, Berlin (1996).
  42. 42.
    Levallois, J., Coeurjolly, D., Lachaud, J.: Parameter-free and multigrid convergent digital curvature estimators. In: Proceedings of Discrete Geometry for Computer Imagery—18th IAPR International Conference, DGCI 2014, Siena, Italy, September 10–12, 2014, pp. 162–175 (2014)Google Scholar
  43. 43.
    Lévy, B., Zhang, H.: Spectral Mesh Processing. Technical report, SIGGRAPH Asia 2009 courses (2008)Google Scholar
  44. 44.
    Mayer, U.: Numerical solutions for the surface diffusion flow in three space dimensions 20 (2001).
  45. 45.
    Mercat, C.: Discrete Riemann surfaces and the Ising model. Commun. Math. Phys. 218(1), 177–216 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Mercat, C.: Discrete complex structure on surfel surfaces. In: Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science, vol. 4992, pp. 153–164. Springer, Berlin (2008).
  47. 47.
    Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Discrete Differential-Geometry Operators for Triangulated 2-Manifolds, pp. 35–57. Springer, Berlin (2003).
  48. 48.
    Molchanov, S.A.: Diffusion processes and Riemannian geometry. Russ. Math. Surv. 30(1), 1 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Ovsjanikov, M., Corman, E., Bronstein, M., Rodolà, E., Ben-Chen, M., Guibas, L., Chazal, F., Bronstein, A.: Computing and processing correspondences with functional maps. In: ACM SIGGRAPH 2017 Courses, SIGGRAPH’17, pp. 5:1–5:62. ACM, New York, NY, USA (2017).
  50. 50.
    Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Polthier, K.: Computational aspects of discrete minimal surfaces (2002).
  52. 52.
    Polthier, K., Preuss, E.: Identifying vector field singularities using a discrete Hodge decomposition. Vis. Math. 3, 113–134 (2003)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Pottmann, H., Wallner, J., Huang, Q., Yang, Y.: Integral invariants for robust geometry processing. Comput. Aided Geom. Des. 26(1), 37–60 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    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). MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Qin, H., Chen, Y., Wang, Y., Hong, X., Yin, K., Huang, H.: Laplace–Beltrami operator on point clouds based on anisotropic Voronoi diagram. Computer Graphics Forum.
  56. 56.
    Regge, T.: General relativity without coordinates. Il Nuovo Cimento Series 10 19(3), 558–571 (1961).
  57. 57.
    Rosenberg, S.: The Laplacian on a Riemannian Manifold. Cambridge University Press (1997). Cambridge Books OnlineGoogle Scholar
  58. 58.
    Rudin, W.: Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, New York (1976)Google Scholar
  59. 59.
    Taubin, G.: A signal processing approach to fair surface design. In: Proceedings of SIGGRAPH, pp. 351–358 (1995).
  60. 60.
    Taubin, G.: Geometric signal processing on polygonal meshes 4 (2001).
  61. 61.
    Vallet, B., Lévy, B.: Spectral geometry processing with manifold harmonics. Comput. Graph. Forum 27(2), 251–260 (2008). CrossRefGoogle Scholar
  62. 62.
    Varadhan, S.: On the behavior of the fundamental solution of the heat equation with variable coefficients. Commun. Pure Appl. Math. 20(2), 431–455 (1967). MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Wardetzky, M.: Discrete differential operators on polyhedral surfaces—convergence and approximation. Ph.D. thesis, Freie Universität Berlin (2010)Google Scholar
  64. 64.
    Wardetzky, M., Mathur, S., Kaelberer, F., Grinspun, E.: Discrete Laplace operators: no free lunch. Eurographics Symposium on Geometry Processing, pp. 33–37 (2007).
  65. 65.
    Willmore, T.: Riemannian Geometry. Oxford Science Publications, Clarendon Press (1996)Google Scholar
  66. 66.
    Xu, G.: Convergence of discrete Laplace–Beltrami operators over surfaces. Comput. Math. Appl. 48(3), 347–360 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Xu, G.: Discrete Laplace–Beltrami operators and their convergence. Comput. Aided Geom. Des. 21(8), 767–784 (2004). Geometric Modeling and Processing 2004
  68. 68.
    Xu, G.: Convergence analysis of a discretization scheme for Gaussian curvature over triangular surfaces. Comput. Aided Geom. Des. 23(2), 193–207 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Zhang, H.: Discrete combinatorial Laplacian operators for digital geometry processing. In: in SIAM Conference on Geometric Design, 2004, pp. 575–592. Press (2004)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Univ Lyon, CNRS, INSA-Lyon, LIRIS, UMR 5205VilleurbanneFrance
  2. 2.Laboratoire de Mathématiques (LAMA), UMR 5127 CNRSUniversité Savoie Mont BlancChambéryFrance

Personalised recommendations