Bounds on the k-neighborhood for locally uniform sampled surfaces

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Bounds on the k-neighborhood for locally uniform sampled surfaces

Eurographics Association


Eurographics Symposium on Point-Based Graphics (2004) M. Alexa, S. Rusinkiewicz (Editors) Bounds on the k-Neighborhood for Locally Uniformly Sampled Surfaces Mattias Andersson1 Joachim Giesen2† Mark Pauly3 Bettina Speckmann4 1 Department of Computer Science, Lund Institute of Technology, S-22100 Lund, Sweden 2 Department of Computer Science, ETH Zurich, CH-8092 Zurich, Switzerland 3 Computer Science Department, Stanford University, Stanford CA 94305, USA 4 Department of Mathematics and Computer Science, TU Eindhoven, 5600 MB Eindhoven, The Netherlands Abstract Given a locally uniform sample set P of a smooth surface S. We derive upper and lower bounds on the number k of nearest neighbors of a sample point p that have to be chosen from P such that this neighborhood contains all restricted Delaunay neighbors of p. In contrast to the trivial lower bound, the upper bound indicates that a sampling condition that is used in many computational geometry proofs is quite reasonable from a practical point of view. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Curve, sur- face, solid, and object representations 1. Introduction Point primitives have recently become popular as a means of surface representation in computer graphics and geometric modeling. With an abundance of 3D acquisition and sampling methods that create point samples from surfaces, numerous algorithms for di- rect processing and rendering of these data sets have been proposed. These methods typically exploit the structural simplicity of point clouds for compact stor- age [KV03], fast re-sampling [PGK02], and efficient rendering [RL00, BWK02]. In the most general case, 3D acquisition or sam- pling methods create a finite set of points that only sample the 3D position of the underlying surface. However, subsequent processing or rendering algo- rithms [ZPvG01, KV01, PKKG03] require additional geometric information on the surface, such as surface normals or local curvatures, which have to be

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