Approximating polyhedra with spheres for time-critical collision detection
ACM Transactions on Graphics (TOG)
Accurate and efficient unions of balls
Proceedings of the sixteenth annual symposium on Computational geometry
Delaunay based shape reconstruction from large data
PVG '01 Proceedings of the IEEE 2001 symposium on parallel and large-data visualization and graphics
The Medial axis of a union of balls
Computational Geometry: Theory and Applications
Adaptive medial-axis approximation for sphere-tree construction
ACM Transactions on Graphics (TOG)
Feature-Sensitive 3D Shape Matching
CGI '04 Proceedings of the Computer Graphics International
Efficient and robust computation of an approximated medial axis
SM '04 Proceedings of the ninth ACM symposium on Solid modeling and applications
Poisson surface reconstruction
SGP '06 Proceedings of the fourth Eurographics symposium on Geometry processing
Finding the Homology of Submanifolds with High Confidence from Random Samples
Discrete & Computational Geometry
Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees
Computational Geometry: Theory and Applications
Reconstruction Using Witness Complexes
Discrete & Computational Geometry
A Sampling Theory for Compact Sets in Euclidean Space
Discrete & Computational Geometry
The power crust, unions of balls, and the medial axis transform
Computational Geometry: Theory and Applications
Natural neighbor coordinates of points on a surface
Computational Geometry: Theory and Applications
Watertight scenes from urban LiDar and planar surfaces
SGP '13 Proceedings of the Eleventh Eurographics/ACMSIGGRAPH Symposium on Geometry Processing
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For a surface F in 3-space that is represented by a set S of sample points, we construct a coarse approximating polytope P that uses a subset of S as its vertices and preserves the topology of F. In contrast to surface reconstruction we do not use all the sample points, but we try to use as few points as possible. Such a polytope P is useful as a 'seed polytope' for starting an incremental refinement procedure to generate better and better approximations of F based on interpolating subdivision surfaces or e.g. Bézier patches. Our algorithm starts from an r-sample S of F. Based on S, a set of surface covering balls with maximal radii is calculated such that the topology is retained. From the weighted a-shape of a proper subset of these highly overlapping surface balls we get the desired polytope. As there is a rather large range for the possible radii for the surface balls, the method can be used to construct triangular surfaces from point clouds in a scalable manner. We also briefly sketch how to combine parts of our algorithm with existing medial axis algorithms for balls, in order to compute stable medial axis approximations with scalable level of detail.