Three-dimensional alpha shapes
ACM Transactions on Graphics (TOG)
Surface reconstruction by Voronoi filtering
Proceedings of the fourteenth annual symposium on Computational geometry
Computational topology: ambient isotopic approximation of 2-manifolds
Theoretical Computer Science - Topology in computer science
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
Manifold reconstruction from point samples
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Graphical Models
Surface reconstruction from noisy point clouds
SGP '05 Proceedings of the third Eurographics symposium on Geometry processing
A sampling theory for compact sets in Euclidean space
Proceedings of the twenty-second annual symposium on Computational geometry
Weak witnesses for Delaunay triangulations of submanifolds
Proceedings of the 2007 ACM symposium on Solid and physical modeling
Manifold reconstruction in arbitrary dimensions using witness complexes
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Reconstruction using witness complexes
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Complexity of Delaunay triangulation for points on lower-dimensional polyhedra
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
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
Towards persistence-based reconstruction in euclidean spaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Cut locus and topology from surface point data
Proceedings of the twenty-fifth annual symposium on Computational geometry
Manifold homotopy via the flow complex
SGP '09 Proceedings of the Symposium on Geometry Processing
Approximating loops in a shortest homology basis from point data
Proceedings of the twenty-sixth annual symposium on Computational geometry
Adaptive surface reconstruction based on implicit PHT-splines
Proceedings of the 14th ACM Symposium on Solid and Physical Modeling
Parallel and adaptive surface reconstruction based on implicit PHT-splines
Computer Aided Geometric Design
New Bounds on the Size of Optimal Meshes
Computer Graphics Forum
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Given a smooth compact codimension one submanifold S of Rk 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 & al. and Dey & al. 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. Our results allow to consider balls of different radii. For the first approximation condition, we also prove that a subset (known as the λ medial axis) of the medial axis of Rk\K is homotopy equivalent to the medial axis of S. We obtain similar results for smooth compact submanifolds S of Rk of any codimension.