David Cohen-Steiner
ABSTRACT
We address the problem of curvature estimation from sampled smooth surfaces. Building upon the theory of normal cycles, we derive a definition of the curvature tensor for polyhedral surfaces. This definition consists in a very simple and new formula. When applied to a polyhedral approximation of a smooth surface, it yields an efficient and reliable curvature estimation algorithm. Moreover, we bound the difference between the estimated curvature and the one of the smooth surface in the case of restricted Delaunay triangulations.
- Nina Amenta, Sunghee Choi, Tamal Dey and Naveen Leekha. A simple algorithm for homeomorphic surface reconstruction. Proc. 16th Annu. ACM Sympos. Comput. Geom., pages 213--222, 2000. Google Scholar
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- Nina Amenta and Marshall Bern. Surface reconstruction by Voronoi filtering. Discrete Comput. Geom., 22(4):481--504, 1999.Google ScholarCross Ref
- S.Funke and E.A.Ramos Smooth-Surface Reconstruction in Near-Linear Time, to appear in SODA 2002. Google ScholarDigital Library
- H. Edelsbrunner and N. R. Shah. Triangulating topological spaces. Int. J. on Comp. Geom., 7:365--378, 1997.Google ScholarCross Ref
- Herbert Edelsbrunner, John Harer, and Afra Zomorodian. Hierarchical Morse complexes for piecewise linear 2-manifolds. In Proc. 17th Annu. ACM Sympos. Comput. Geom., pages 70--79, 2001. Google ScholarDigital Library
- Th. Banchoff, Critical points and curvature for embedded polyhedra, J. Diff. Geom 1 (1967) 245--256.Google ScholarCross Ref
- M. Berger, B. Gostiaux, Géométrie différentielle : variétés, courbes et surfaces, Presses Universitaires de France.Google Scholar
- H.Cartan, Cours de calcul différentiel, Hermann.Google Scholar
- H. Federer, Curvature measure theory, Trans. Amer. Math. Soc 93 (1959) 418--491.Google ScholarCross Ref
- H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1983.Google Scholar
- J. Fu, Convergence of curvatures in secant approximations, J.Differential Geometry 37 (1993) 177--190.Google ScholarCross Ref
- F. Morgan, Geometric measure theory, Acad. Press, INC. 1987.Google Scholar
- J. Steiner, Jber Preuss. Akad. Wiss. 114--118,(1840). In Gesammelte Werke, vol 2, New York, Chelsea 1971.Google Scholar
- P. Wintgen, Normal cycle and integral curvature for polyhedra in Riemmannian manifolds, Differential Geometry (Gy. Soos and J. Szenthe, eds.), North-Holland, Amsterdam, 1982.Google Scholar
- M. Zahle, Integral and current representations of Federer's curvature measures, Arch. Math. (Basel) 46, (1986), 557--567.Google ScholarCross Ref
- J.M. Morvan, On generalized curvatures, in preparation. Google ScholarDigital Library
- S. Petitjean, A survey of methods for recovering quadrics in triangle meshes, accepted.Google Scholar
- M. Desbrun, M. Meyer, P. Schroder and A. Barr Discrete differential-geometry operators in nD, preprint, the Caltech Multi-Res Modeling Group.Google Scholar
- D. Meek and D. Walton, On surface normal and Gaussian curvature approximation given data sampled from a smooth surface, Computer-Aided Geometric Design 17, 521--543. Google ScholarDigital Library
- G. Taubin, Estimating the Tensor of Curvature of a Surface from a Polyhedral Approximation, Fifth International Conference on Computer Vision (ICCV'95). Google ScholarDigital Library
- P. Alliez, D. Cohen-Steiner, M. Desbrun, O. Devillers and B. Lévy, Anisotropic Polygonal Remeshing, to appear in SIGGRAPH 2003. Google ScholarDigital Library
Index Terms
- Restricted delaunay triangulations and normal cycle
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