Three-dimensional alpha shapes
ACM Transactions on Graphics (TOG)
Stability and Finiteness Properties of Medial Axis and Skeleton
Journal of Dynamical and Control Systems
Provable surface reconstruction from noisy samples
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Graphical Models
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
Stability and Computation of Topological Invariants of Solids in ${\Bbb R}^n$
Discrete & Computational Geometry
Surface reconstruction from noisy point clouds
SGP '05 Proceedings of the third Eurographics symposium on Geometry processing
Finding the Homology of Submanifolds with High Confidence from Random Samples
Discrete & Computational Geometry
Recovering structure from r-sampled objects
SGP '09 Proceedings of the Symposium on Geometry Processing
Manifold reconstruction using tangential Delaunay complexes
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
Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes
Proceedings of the twenty-seventh annual symposium on Computational geometry
Boundary of a non-uniform point cloud for reconstruction: extended abstract
Proceedings of the twenty-seventh annual symposium on Computational geometry
A Topological View of Unsupervised Learning from Noisy Data
SIAM Journal on Computing
Preserving geometric properties in reconstructing regions from internal and nearby points
Computational Geometry: Theory and Applications
The Journal of Machine Learning Research
Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes
Computational Geometry: Theory and Applications
Homological reconstruction and simplification in R3
Proceedings of the twenty-ninth annual symposium on Computational geometry
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Given a smooth compact codimension one submanifold S of R^k and a compact approximation K of S, we prove that it is possible to reconstruct S and to approximate the medial axis of S with topological guarantees using unions of balls centered on K. We consider two notions of noisy-approximation that generalize sampling conditions introduced by Amenta et al. and Dey et al. The first one generalizes uniform sampling based on the minimum value of the local feature size. The second one generalizes non-uniform sampling based on the local feature size function of S. The density and noise of the approximation are bounded by a constant times the local feature size function. This constant does not depend on the surface S. Our results are based upon critical point theory for distance functions. For the two approximation conditions, we prove that the connected components of the boundary of unions of balls centered on K are isotopic to S. We consider using both balls of uniform radius and also balls whose radii vary with the local level of detail of the manifold. For the first approximation condition, we prove that a subset (known as the @l-medial axis) of the medial axis of R^k@?K is homotopy equivalent to the medial axis of S. Our results generalize to smooth compact submanifolds S of R^k of any codimension.