Guaranteed-quality mesh generation for curved surfaces
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Learning smooth objects by probing
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Graphical Models
Provably good sampling and meshing of surfaces
Graphical Models - Solid modeling theory and applications
A sampling theory for compact sets in Euclidean space
Proceedings of the twenty-second annual symposium on Computational geometry
A flexible framework for surface reconstruction from large point sets
Computers and Graphics
Manifold reconstruction in arbitrary dimensions using witness complexes
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Reconstruction using witness complexes
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Delaunay refinement for piecewise smooth complexes
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Geodesic Delaunay triangulation and witness complex in the plane
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Complete subdivision algorithms, II: isotopic meshing of singular algebraic curves
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
From Segmented Images to Good Quality Meshes Using Delaunay Refinement
Emerging Trends in Visual Computing
Provably correct reconstruction of surfaces from sparse noisy samples
Pattern Recognition
Manifold homotopy via the flow complex
SGP '09 Proceedings of the Symposium on Geometry Processing
Isotropic remeshing with fast and exact computation of Restricted Voronoi Diagram
SGP '09 Proceedings of the Symposium on Geometry Processing
Lp Centroidal Voronoi Tessellation and its applications
ACM SIGGRAPH 2010 papers
Optimal reconstruction might be hard
Proceedings of the twenty-sixth annual symposium on Computational geometry
Geodesic delaunay triangulations in bounded planar domains
ACM Transactions on Algorithms (TALG)
Multiscale acquisition and presentation of very large artifacts: The case of portalada
Journal on Computing and Cultural Heritage (JOCCH)
Complete subdivision algorithms, II: Isotopic meshing of singular algebraic curves
Journal of Symbolic Computation
Shape reconstruction from unorganized set of points
ICIAR'10 Proceedings of the 7th international conference on Image Analysis and Recognition - Volume Part I
Localized delaunay refinement for piecewise-smooth complexes
Proceedings of the twenty-ninth annual symposium on Computational geometry
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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.