Power diagrams: properties, algorithms and applications
SIAM Journal on Computing
Journal of the ACM (JACM)
Smooth-surface reconstruction in near-linear time
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Efficient Simplicial Reconstructions of Manifolds from Their Samples
IEEE Transactions on Pattern Analysis and Machine Intelligence
Shape dimension and intrinsic metric from samples of manifolds with high co-dimension
Proceedings of the nineteenth annual symposium on Computational geometry
A greedy Delaunay-based surface reconstruction algorithm
The Visual Computer: International Journal of Computer Graphics
Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment
SIAM Journal on Scientific Computing
Star splaying: an algorithm for repairing delaunay triangulations and convex hulls
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
IEEE Transactions on Pattern Analysis and Machine Intelligence
Curve and Surface Reconstruction: Algorithms with Mathematical Analysis (Cambridge Monographs on Applied and Computational Mathematics)
Manifold reconstruction in arbitrary dimensions using witness complexes
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
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
Towards persistence-based reconstruction in euclidean spaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Locally uniform anisotropic meshing
Proceedings of the twenty-fourth annual symposium on Computational geometry
Discrete laplace operator on meshed surfaces
Proceedings of the twenty-fourth 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
A fast and simple surface reconstruction algorithm
Proceedings of the twenty-eighth annual symposium on Computational geometry
Visible neighborhood graph of point clouds
Graphical Models
The Journal of Machine Learning Research
Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes
Computational Geometry: Theory and Applications
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We give a provably correct algorithm to reconstruct a k-dimensional manifold embedded in d-dimensional Euclidean space. Input to our algorithm is a point sample coming from an unknown manifold. Our approach is based on two main ideas : the notion of tangential Delaunay complex defined in [6,19,20], and the technique of sliver removal by weighting the sample points [13]. Differently from previous methods, we do not construct any subdivision of the embedding d-dimensional space. As a result, the running time of our algorithm depends only linearly on the extrinsic dimension d while it depends quadratically on the size of the input sample, and exponentially on the intrinsic dimension k. To the best of our knowledge, this is the first certified algorithm for manifold reconstruction whose complexity depends linearly on the ambient dimension. We also prove that for a dense enough sample the output of our algorithm is isotopic to the manifold and a close geometric approximation of the manifold.