The crust and the &Bgr;-Skeleton: combinatorial curve reconstruction
Graphical Models and Image Processing
The flow complex: a data structure for geometric modeling
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
A geometric convection approach of 3-D reconstruction
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Loops in Reeb Graphs of 2-Manifolds
Discrete & Computational Geometry
Smooth surface reconstruction via natural neighbour interpolation of distance functions
Computational Geometry: Theory and Applications
The power crust, unions of balls, and the medial axis transform
Computational Geometry: Theory and Applications
Identifying flat and tubular regions of a shape by unstable manifolds
Proceedings of the 2006 ACM symposium on Solid and physical modeling
Topology guaranteeing manifold reconstruction using distance function to noisy data
Proceedings of the twenty-second annual symposium on Computational geometry
A sampling theory for compact sets in Euclidean space
Proceedings of the twenty-second annual symposium on Computational geometry
Medial axis approximation and unstable flow complex
Proceedings of the twenty-second annual symposium on Computational geometry
Geometric and topological guarantees for the WRAP reconstruction algorithm
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
The flow complex: A data structure for geometric modeling
Computational Geometry: Theory and Applications
Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees
Computational Geometry: Theory and Applications
Multi-component heart reconstruction from volumetric imaging
Proceedings of the 2008 ACM symposium on Solid and physical modeling
Robust construction of the three-dimensional flow complex
Proceedings of the twenty-fourth annual symposium on Computational geometry
Normal cone approximation and offset shape isotopy
Computational Geometry: Theory and Applications
Surface reconstruction from point clouds by transforming the medial scaffold
Computer Vision and Image Understanding
Manifold homotopy via the flow complex
SGP '09 Proceedings of the Symposium on Geometry Processing
Optimal reconstruction might be hard
Proceedings of the twenty-sixth annual symposium on Computational geometry
Reconstructing shapes with guarantees by unions of convex sets
Proceedings of the twenty-sixth annual symposium on Computational geometry
Quality tetrahedral mesh generation for macromolecules
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
SPBG'05 Proceedings of the Second Eurographics / IEEE VGTC conference on Point-Based Graphics
A parallel algorithm for computing the flow complex
Proceedings of the twenty-ninth annual symposium on Computational geometry
SMI 2013: Minimizing edge length to connect sparsely sampled unstructured point sets
Computers and Graphics
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The distance function to surfaces in three dimensions plays a key role in many geometric modeling applications such as medial axis approximations, surface reconstructions, offset computations, feature extractions and others. In most cases, the distance function induced by the surface is approximated by a discrete distance function induced by a discrete sample of the surface. The critical points of the distance function determine the topology of the set inducing the function. However, no earlier theoretical result has linked the critical points of the distance to a sampling of geometric structures to their topological properties. We provide this link by showing that the critical points of the distance function induced by a discrete sample of a surface either lie very close to the surface or near its medial axis and this closeness is quantified with the sampling density. Based on this result, we provide a new flow-complex-based surface reconstruction algorithm that, given a tight ε-sampling of a surface, approximates the surface geometrically, both in Hausdorff distance and normals, and captures its topology.