Abstract
In geometric modeling and processing, computer graphics and computer vision, smooth surfaces are approximated by discrete triangular meshes reconstructed from sample points on the surface. A fundamental problem is to design rigorous algorithms to guarantee the geometric approximation accuracy by controlling the sampling density.
This theoretic work gives explicit formula to the bounds of Hausdorff distance, normal distance and Riemannian metric distortion between the smooth surface and the discrete mesh in terms of principle curvature and the radii of geodesic circum-circle of the triangles. These formula can be directly applied to design sampling density for data acquisition and surface reconstructions.
Furthermore, we prove the meshes induced from the Delaunay triangulations of the dense samples on a smooth surface are convergent to the smooth surface under both Hausdorff distance and normal fields. The Riemannian metrics and the Laplace-Beltrami operators on the meshes are also convergent. These theoretic results lay down the theoretic foundation for a broad class of reconstruction and approximation algorithms in geometric modeling and processing.
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References
Hoppe, H., DeRose, T., Duchamp, T., McDonald, J.A., Stuetzle, W.: Surface reconstruction from unorganized points. In: SIGGRAPH, pp. 71–78 (1992)
Eck, M., Hoppe, H.: Automatic reconstruction of b-spline surfaces of arbitrary topological type. In: SIGGRAPH, pp. 325–334 (1996)
Amenta, N., Bern, M.W., Kamvysselis, M.: A new voronoi-based surface reconstruction algorithm. In: SIGGRAPH, pp. 415–421 (1998)
Amenta, N., Bern, M.W.: Surface reconstruction by voronoi filtering. Discrete & Computational Geometry 22, 481–504 (1999)
Amenta, N., Choi, S., Dey, T.K., Leekha, N.: A simple algorithm for homeomorphic surface reconstruction. Int. J. Comput. Geometry Appl. 12, 125–141 (2002)
Bajaj, C.L., Bernardini, F., Xu, G.: Automatic reconstruction of surfaces and scalar fields from 3d scans. In: SIGGRAPH, pp. 109–118 (1995)
Bernardini, F., Bajaj, C.L.: Sampling and reconstructing manifolds using alpha-shapes. In: CCCG (1997)
Ju, T., Losasso, F., Schaefer, S., Warren, J.D.: Dual contouring of hermite data. In: SIGGRAPH, pp. 339–346 (2002)
Floater, M.S., Reimers, M.: Meshless parameterization and surface reconstruction. Computer Aided Geometric Design 18, 77–92 (2001)
Benkö, P., Martin, R.R., Várady, T.: Algorithms for reverse engineering boundary representation models. Computer-Aided Design 33, 839–851 (2001)
He, Y., Qin, H.: Surface reconstruction with triangular b-splines. In: GMP, pp. 279–290 (2004)
Hoppe, H.: Progressive meshes. In: SIGGRAPH, pp. 99–108 (1996)
Leibon, G., Letscher, D.: Delaunay triangulations and voronoi diagrams for riemannian manifolds. In: Symposium on Computational Geometry, pp. 341–349 (2000)
Elber, G.: Error bounded piecewise linear approximation of freeform surfaces. Computer Aided Design 28, 51–57 (1996)
Morvan, J.M., Thibert, B.: On the approximation of a smooth surface with a triangulated mesh. Computational Geometry Theory and Application 23, 337–352 (2002)
Morvan, J.M., Thibert, B.: Approximation of the normal vector field and the area of a smooth surface. Discrete and Computational Geometr 32, 383–400 (2004)
Surazhsky, V., Surazhsky, T., Kirsanov, D., Gortler, S.J., Hoppe, H.: Fast exact and approximate geodesics on meshes. ACM Trans. Graph. 24, 553–560 (2005)
Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Dodgson, N.A., Floater, M.S., Sabin, M.A. (eds.) Advances in multiresolution for geometric modelling, pp. 157–186. Springer, Heidelberg (2005)
Ben-Chen, M., Gotsman, C.: On the optimality of spectral compression of mesh data. ACM Trans. Graph. 24, 60–80 (2005)
Hildebrandt, K., Polthier, K., Wardetzky, M.: On the convergence of metric and geometric properties of polyhedral surfaces (submitted, 2005)
Federer, H.: Curvature measures. Transactions of the American Mathematical Society 93, 418–491 (1959)
Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. International Press, Cambridge (1994)
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Dai, J., Luo, W., Yau, ST., Gu, X.D. (2006). Geometric Accuracy Analysis for Discrete Surface Approximation. In: Kim, MS., Shimada, K. (eds) Geometric Modeling and Processing - GMP 2006. GMP 2006. Lecture Notes in Computer Science, vol 4077. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11802914_5
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DOI: https://doi.org/10.1007/11802914_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-36711-6
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