Abstract
The goal of a multilevel simplification method is to produce different levels of refinement of a mesh, reducing the resolution (total number of faces), while preserving the original topology and a good approximation to the original geometry. A new approach to simplification based on the evolution of surfaces under p-Laplacian flow is presented. Such an evolution provides a natural geometric clustering process where the spatial effect of the p-Laplacian allows for identifying suitable regions that need to be simplified. The concrete scheme is a multiresolution framework composed, at each simplification level, of a spatial clustering diffusion flow to determine the potential candidates for deletion, followed by an incremental decimation process to update the mesh vertex locations in order to decrease the overall resolution. Numerical results show the effectiveness of our strategy in multilevel simplification of different models with different complexities, in particular for models characterized by sharp features and flat parts.
Similar content being viewed by others
References
Cignoni, P., Montani, C., Scopigno, R.: A comparison of mesh simplification algorithms. Comput. Graph. 22(1), 37–54 (1997)
Luebke, D., Watson, B., Cohen, J., Reddy, M., Varshney, A.: Level of Detail for 3D Graphics. Elsevier, Amsterdam (2002)
Garland, M., Heckbert, P.S.: Surface simplification using quadric error metrics. In: Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’97, New York, NY, USA, pp. 209–216. ACM Press/Addison-Wesley, New York (1997)
Botsch, M., Kobbelt, L., Pauly, M., Alliez, P., Levy, B.: Polygon Mesh Processing. AK Peters, Wellesley (2010)
Rossignac, J., Borrel, P.: Multi-resolution 3D approximation for rendering complex scenes. In: Falcidieno, B., Kunii, T.L. (eds.) Geometric Modeling in Computer Graphics: Graphics: Methods and Applications, pp. 455–465. Springer, Berlin (1993)
Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., Stuetzle, W.: Mesh optimization. In: Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’93, pp. 19–26 (1993)
Bae, E., Weickert, J.: Partial differential equations for interpolation and compression of surfaces. In: Proceedings of the 7th International Conference on Mathematical Methods for Curves and Surfaces, MMCS’08, Berlin, Heidelberg, pp. 1–14. Springer, Berlin (2010)
Schroeder, W., Zarge, J., Lorensen, W.: Decimation of triangle meshes. In: Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques. ACM SIGGRAPH Computer Graphics, vol. 26(1), pp. 65–70 (1992)
Hoppe, H.: Progressive meshes. In: Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’96, pp. 99–108 (1996)
Guskov, I., Sweldens, W., Schröder, P.: Multiresolution signal processing for meshes. In: Computer Graphics Proceedings (SIGGRAPH 99), pp. 325–334 (1999)
Hoppe, H.: New quadric metric for simplifying meshes with appearance attributes. In: Proceedings of the 10th IEEE Visualization 1999 Conference (VIS ’99), VISUALIZATION ’99, pp. 59–66 (1999)
Kim, S., Kim, C., Levin, D.: Surface simplification using a discrete curvature norm. Comput. Graph. 26(5), 657–663 (2002)
Borouchaki, H., Frey, P.: Simplification of surface mesh using Hausdorff envelope. Comput. Methods Appl. Mech. Eng. 194(48–49), 4864–4884 (2005)
Oberman, A.: Finite difference methods for the infinity Laplace and p-Laplace equations. Preprint, pp. 1–14, July 2011
Chan, T., Chen, K.: An optimization-based multilevel algorithm for total variation image denoising. Multiscale Model. Simul. 5(2), 615–645 (2006)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)
Li, Y., Santosa, F.: A computational algorithm for minimizing total variation in image restoration. IEEE Trans. Image Process. 5(6), 987–995 (1996)
Ng, M.K., Shen, H., Lam, E.Y., Zhang, L.: A total variation regularization based super-resolution reconstruction algorithm for digital video. EURASIP J. Adv. Signal Process. (2007). doi:10.1155/2007/74585
Huang, Y., Li, R., Liu, W.: Preconditioned descent algorithms for p-Laplacian. J. Sci. Comput. 32(2), 343–371 (2007)
Ainsworth, M., Kay, D.: The approximation theory for the p-version finite element method and application to non-linear elliptic PDEs. Numer. Math. 82(3), 351–388 (1999)
Liu, W., Yan, N.: On quasi-norm interpolation error estimation and a posteriori error estimates for p-Laplacian. SIAM J. Numer. Anal. 40(5), 1780–1895 (2002)
Zhou, D., Scholkopf, B.: A regularization framework for learning from graph data. In: ICML Workshop on Statistical Relational Learning, pp. 132–137 (2004). Other Fields
Zhou, D., Schölkopf, B.: Regularization on discrete spaces. In: Pattern Recognition, pp. 361–368. Springer, Berlin (2005)
Bougleux, S., Elmoataz, A., Melkemi, M.: Discrete regularization on weighted graphs for image and mesh filtering. In: International Conference on Scale Space and Variational Methods in Computer Vision (SSVM), pp. 128–139 (2007)
Elmoataz, A., Lezoray, O., Bougleux, S.: Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing. IEEE Trans. Image Process. 17(7), 1047–1060 (2008)
Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2, 15–36 (1993)
Meyer, M., Desbrun, M., Schröder, P., Barr, A.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Visualization and Mathematics III, pp. 35–57. Springer, Berlin (2003)
Desbrun, M., Meyer, M., Schröder, P., Barr, A.: Implicit fairing of irregular meshes using diffusion and curvature flow. In: SIGGRAPH ’99 Conference Proceedings, pp. 317–324 (1999)
Schneider, R., Kobbelt, L.: Geometric fairing of irregular meshes for free-form surface design. Comput. Aided Geom. Des. 18, 359–379 (2001)
Morigi, S., Rucci, M., Sgallari, F.: Nonlocal surface fairing. In: Third International Conference on Scale Space and Variational Methods in Computer Vision (SSVM), pp. 38–49 (2011)
Cignoni, P., Callieri, M., Corsini, M., Dellepiane, M., Ganovelli, F., Ranzuglia, G.: MeshLab an open-source mesh processing tool. In: Sixth Eurographics Italian Chapter, pp. 129–136 (2008)
Cignoni, P., Rocchini, C., Scopigno, R.: Metro: measuring error on simplified surfaces. Comput. Graph. Forum 17(2), 167–174 (1998)
Cunderlik, R., Mikula, K., Tunega, M.: Nonlinear diffusion filtering of data on the earth’s surface. J. Geod. 87(2), 143–160 (2013)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Morigi, S., Rucci, M. Multilevel mesh simplification. Vis Comput 30, 479–492 (2014). https://doi.org/10.1007/s00371-013-0873-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00371-013-0873-6