ABSTRACT
In the last decade, a great deal of work has been devoted to the elaboration of a sampling theory for smooth surfaces. The goal was to ensure a good reconstruction of a given surface S from a finite subset E of S. The sampling conditions proposed so far offer guarantees provided that E is sufficiently dense with respect to the local feature size of S, which can be true only if S is smooth since the local feature size vanishes at singular points.In this paper, we introduce a new measurable quantity, called the Lipschitz radius, which plays a role similar to that of the local feature size in the smooth setting, but which is well-defined and positive on a much larger class of shapes. Specifically, it characterizes the class of Lipschitz surfaces, which includes in particular all piecewise smooth surfaces such that the normal deviation is not too large around singular points.Our main result is that, if S is a Lipschitz surface and E is a sample of S such that any point of S is at distance less than a fraction of the Lipschitz radius of S, then we obtain similar guarantees as in the smooth setting. More precisely, we show that the Delaunay triangulation of E restricted to S is a 2-manifold isotopic to S lying at bounded Hausdorff distance from S, provided that its facets are not too skinny.We further extend this result to the case of loose samples. As an application, the Delaunay refinement algorithm we proved correct for smooth surfaces works as well and comes with similar guarantees when applied to Lipschitz surfaces.
- N. Amenta and M. Bern. Surface reconstruction by Voronoi filtering. Discrete Comput. Geom., 22(4):481--504, 1999.Google ScholarCross Ref
- N. Amenta, S. Choi, T. K. Dey, and N. Leekha. A simple algorithm for homeomorphic surface reconstruction. Internat. Journal of Comput. Geom. and Applications, 12:125--141, 2002.Google ScholarCross Ref
- M. Berger and B. Costiaux. Differential Geometry: Manifolds, Curves, and Surfaces, volume 115 of Graduate Texts in Mathematics Series. Springer, 1988. 474 pages.Google Scholar
- J.-D. Boissonnat, L. J. Guibas, and S. Oudot. Learning smooth objects by probing. In Proc. 21st Annu. ACM. Sympos. Comput. Geom., pages 198--207, 2005. Google ScholarDigital Library
- J.-D. Boissonnat and S. Oudot. Provably good sampling and meshing of surfaces. Graphical Models, 67(5):405--451, September 2005. Google ScholarDigital Library
- F. Chazal, D. Cohen-Steiner, and A. Lieutier. A sampling theory for compacts in Euclidean space. In Proc. 22nd Annu. ACM Sympos. Comput. Geom., 2006. Google ScholarDigital Library
- F. Chazal and A. Lieutier. Weak feature size and persistent homology: Computing homology of solids in Rn from noisy data samples. Technical Report 378, Institut de Mathématiques de Bourgogne, 2004. Partially published in {9}.Google Scholar
- F. Chazal and A. Lieutier. The λ-medial axis. Graphical Models, 67(4):304--331, July 2005. Google ScholarDigital Library
- F. Chazal and A. Lieutier. Weak feature size and persistent homology: Computing homology of solids in Rn from noisy data samples. In Proc. 21st Annual ACM Symposium on Computational Geometry, pages 255--262, 2005. Google ScholarDigital Library
- L. P. Chew. Guaranteed-quality mesh generation for curved surfaces. In Proc. 9th Annu. ACM Sympos. Comput. Geom., pages 274--280, 1993. Google ScholarDigital Library
- F. H. Clarke. Optimization and Nonsmooth Analysis. Classics in applied mathematics. SIAM, 1990.Google Scholar
- D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. In Proc. 21st Annual ACM Symposium on Computational Geometry, pages 263--271, 2005. Google ScholarDigital Library
- T. K. Dey, G. Li, and T. Ray. Polygonal surface remeshing with {Delaunay refinement. In Proc. 14th Internat. Meshing Roundtable, 2005.Google ScholarCross Ref
- H. Federer. Geometric Measure Theory. Classics in Mathematics. Springer-Verlag, 1996. Reprint of the 1969 ed.Google Scholar
- M. W. Hirsch. Differential Topology. Springer-Verlag, New York, NY, 1976.Google Scholar
- A. Lieutier. Any open bounded subset of Rn has the same homotopy type as it medial axis. Computer-Aided Design, 36(11):1029--1046, September 2004.Google ScholarCross Ref
- J. Nečas. Les méthodes directes en théorie des équations elliptiques. Masson, 1967.Google Scholar
- S. Oudot. Sampling and Meshing Surfaces with Guarantees. Thèse de doctorat en sciences, ècole Polytechnique, Palaiseau, France, 2005. Preprint available at ftp://ftp-sop.inria.fr/geometrica/soudot/preprints/thesis.pdf.Google Scholar
- S. Oudot, L. Rineau, and M. Yvinec. Meshing volumes bounded by smooth surfaces. In Proc. 14th Internat. Meshing Roundtable, pages 203--219, 2005.Google ScholarCross Ref
Index Terms
- Provably good sampling and meshing of Lipschitz surfaces
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