A Sampling Theory for Compact Sets in Euclidean Space

We introduce a parameterized notion of feature size that interpolates between the minimum of the local feature size, and the recently introduced weak feature size. Based on this notion of feature size, we propose sampling conditions that apply to noisy samplings of general compact sets in euclidean space. These conditions are sufficient to ensure the topological correctness of a reconstruction given by an offset of the sampling. Our approach also yields new stability results for medial axes, critical points and critical values of distance functions.

[1]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[2]  Tamal K. Dey,et al.  Reconstruction curves with sharp corners , 2000, SCG '00.

[3]  Jean-Luc Starck,et al.  Analysis of the Spatial Distribution of Galaxies by Multiscale Methods , 2005, EURASIP J. Adv. Signal Process..

[4]  Jeff Cheeger,et al.  Critical points of distance functions and applications to geometry , 1991 .

[5]  Raphaëlle Chaine.  A geometric convection approach of 3-D reconstruction , 2003 .

[6]  James F. O'Brien,et al.  Spectral surface reconstruction from noisy point clouds , 2004, SGP '04.

[7]  Joachim Giesen,et al.  The flow complex: a data structure for geometric modeling , 2003, SODA '03.

[8]  J. Fu,et al.  Tubular neighborhoods in Euclidean spaces , 1985 .

[9]  Marshall W. Bern,et al.  Surface Reconstruction by Voronoi Filtering , 1998, SCG '98.

[10]  David Cohen-Steiner,et al.  Stability of Persistence Diagrams , 2005, Discret. Comput. Geom..

[11]  Karsten Grove,et al.  A generalized sphere theorem , 1977 .

[12]  Sunghee Choi,et al.  A simple algorithm for homeomorphic surface reconstruction , 2000, SCG '00.

[13]  Tamal K. Dey,et al.  Provable surface reconstruction from noisy samples , 2004, SCG '04.

[14]  Tamal K. Dey,et al.  Critical points of the distance to an epsilon-sampling of a surface and flow-complex-based surface reconstruction , 2005, Symposium on Computational Geometry.

[15]  Jim Ruppert,et al.  A Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh Generation , 1995, J. Algorithms.

[16]  Frédéric Chazal,et al.  Weak feature size and persistent homology: computing homology of solids in Rn from noisy data samples , 2005, SCG.

[17]  J. Boissonnat,et al.  Provably good sampling and meshing of Lipschitz surfaces , 2006, SCG '06.

[18]  Herbert Edelsbrunner,et al.  Three-dimensional alpha shapes , 1992, VVS.

[19]  F. Clarke.  Generalized gradients and applications , 1975 .

[20]  Tamal K. Dey,et al.  Reconstructing curves with sharp corners , 2001, Comput. Geom..

[21]  André Lieutier,et al.  Any open bounded subset of Rn has the same homotopy type as its medial axis , 2004, Comput. Aided Des..

[22]  J. Meiss,et al.  Computing connectedness: disconnectedness and discreteness , 2000 .

[23]  F. Chazal,et al.  Stability and Finiteness Properties of Medial Axis and Skeleton , 2004 .

[24]  Dinesh Manocha,et al.  Efficient computation of a simplified medial axis , 2003, SM '03.

[25]  F. Chazal,et al.  The λ-medial axis , 2005 .

[26]  K. Grove.  Critical point theory for distance functions , 1993 .

[27]  Stephen Smale,et al.  Finding the Homology of Submanifolds with High Confidence from Random Samples , 2008, Discret. Comput. Geom..

[28]  Tamal K. Dey,et al.  Approximate medial axis as a voronoi subcomplex , 2002, SMA '02.

[29]  Frédéric Chazal,et al.  Topology guaranteeing manifold reconstruction using distance function to noisy data , 2006, SCG '06.

[30]  Luiz Velho,et al.  Surface reconstruction from noisy point clouds , 2005, SGP '05.