Abstract
The Laplacian approach, when applied to mesh smoothing leads in many cases to convergence problems. It also leads to shrinking of the mesh. In this work, the authors reformulate the mesh smoothing problem as an optimisation one. This approach gives the means of controlling the steps to assure monotonic convergence. Furthermore, a new optimisation function is proposed that reduces the shrinking effect of the method. Examples are given to illustrate the properties of the proposed approches.
This work has been partially supported by the “ANR BLAN07-2_184378” project.
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References
Taubin, G.: A signal processing approach to fair surface design. In: Computer Graphics Proceedings, Annual Conference Series, pp. 351–358 (1995)
Vollmer, J., Mencl, R., Muller, H.: Improved laplacian smoothing of noisy surface meshes. Computer Graphics Forum 18(3), 131–138 (1999)
Parthasarathy, V., Kodiyalam, S.: A constrained optimization approach to finite element mesh smoothing. Finite Elements in Analysis and Design 9, 309–320 (1991)
Freitag, L.: On combining laplacian and optimization-based mesh smoothing techniques. In: Joint ASME, ASCE, SES symposium on engineering mechanics in manufacturing processes and materials processing, pp. 37–43 (1997)
Amenta, N., Bern, M., Eppstein, D.: Optimal point placement for mesh smoothing. Journal of Algorithms 30, 302–322 (1999)
Bank, R., Smith, R.: Mesh smoothing using a posteriori error estimates. SIAM Journal on Numerical Analysis 34(3), 979–997 (1997)
Freitag, L., Jones, M., Plassmann, P.: A parallel algorithm for mesh smoothing. SIAM Journal on Scientific Computing 20(6), 2023–2040 (1999)
Freitag, L., Knupp, P., Munson, T., Shontz, S.: A comparison of optimization software for mesh shape-quality improvement problems. In: Int. Meshing Roundtable, pp. 29–40 (2002)
Chen, Z., Tristano, J., Kwok, W.: Combined laplacian and optimization-based smoothing for quadratic mixed surface meshes. In: 12th International Meshing Roundtable (2003)
Ji, Z., Liu, L., Wang, G.: A global laplacian smoothing approach with feature preservation. In: Int. Conf. on Computer Aided Design and Computer Graphics, pp. 269–274 (2005)
Nealen, A., Igarashi, T., Sorkine, O., Alexa, M.: Laplacian mesh optimization. In: ACM GRAPHITE, pp. 381–389 (2006)
Bougleux, S., Elmoataz, A., Melkemi, M.: Discrete regularization on weighted graphs for image and mesh filtering. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 128–139. Springer, Heidelberg (2007)
Taubin, G.: Curve and surface smoothing without shrinkage. In: Fifth International Conference on Computer Vision, pp. 852–857 (1995)
Yagou, H., Ohtake, Y., Belyaev, A.: Mesh smoothing via mean and median filtering applied to face normals. In: Procs. Geometric Modeling and Processing, pp. 124–131 (2002)
Djidjev, H.N.: Force-directed methods for smoothing unstructured triangular and tetrahedral meshes. In: Ninth International Meshing Roundtable, pp. 395–406 (2000)
Mezentsev, A.: A generalized graph-theoretic mesh optimization model. In: 13th International Meshing Roundtable, pp. 255–264 (2004)
Ohtake, Y., Belyaev, A.G., Bogaevski, I.A.: Polyhedral surface smoothing with simultaneous mesh regularization. In: Procs. Geometric Modeling and Processing, pp. 229–237 (2000)
Ohtake, Y., Belyaev, A.G., Bogaevski, I.A.: Mesh regularization and adaptive smoothing. Computer-Aided Design 33(11), 789–800 (2001)
Ohtake, Y., Belyaev, A., Seidel, H.: Mesh smoothing by adaptive and anisotropic gaussian filter applied to mesh normals. In: Vision, Modeling, and Visualization, pp. 203–210 (2002)
Tasdizen, T., Whitaker, R., Burchard, P., Osher, S.: Geometric surface smoothing via anisotropic diffusion of normals. In: IEEE Visualization 2002, pp. 125–132 (2002)
Fleishman, S., Drori, I., Cohen-Or, D.: Bilateral mesh denoising. ACM Transactions on Graphics 22(3), 950–953 (2003)
Desbrun, M., Meyer, M., Schröder, P., Barr, A.H.: Implicit fairing of irregular meshes using diffusion and curvature flow. In: 26th annual conference on Computer graphics and interactive techniques, pp. 317–324 (1999)
Zhao, H., Xu, G.: Triangular surface mesh fairing via gaussian curvature flow. Journal of Computational and Applied Mathematics 195(1-2), 300–311 (2006)
Strang, G.: Introduction to Linear Algebra. Wellesley-Cambridge Press (2003)
Sorkine, O.: Differential representations for mesh processing. Computer Graphics Forum 25(4), 789–807 (2006); alt. title: Laplacian mesh processing (Eurographics 2005 presentation)
Chung, F.R.: Spectral Graph Theory. Amer. Mathematical Society, Providence (1997)
Field, D.A.: Laplacian smoothing and delaunay triangulations. Communications in Applied Numerical Methods 4(6), 709–712 (1988)
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Hamam, Y., Couprie, M. (2009). An Optimisation-Based Approach to Mesh Smoothing: Reformulation and Extensions. In: Torsello, A., Escolano, F., Brun, L. (eds) Graph-Based Representations in Pattern Recognition. GbRPR 2009. Lecture Notes in Computer Science, vol 5534. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02124-4_4
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DOI: https://doi.org/10.1007/978-3-642-02124-4_4
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