Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
The crust and the &Bgr;-Skeleton: combinatorial curve reconstruction
Graphical Models and Image Processing
Journal of the ACM (JACM)
Discrete & Computational Geometry
Stability of persistence diagrams
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
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
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
Stability and Computation of Topological Invariants of Solids in ${\Bbb R}^n$
Discrete & Computational Geometry
Reconstruction using witness complexes
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
On the Local Behavior of Spaces of Natural Images
International Journal of Computer Vision
Inferring Local Homology from Sampled Stratified Spaces
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Finding the Homology of Submanifolds with High Confidence from Random Samples
Discrete & Computational Geometry
Topological estimation using witness complexes
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
Analysis of scalar fields over point cloud data
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
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
Gromov-Hausdorff stable signatures for shapes using persistence
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
Topological inference via meshing
Proceedings of the twenty-sixth annual symposium on Computational geometry
Manifold reconstruction using tangential Delaunay complexes
Proceedings of the twenty-sixth annual symposium on Computational geometry
Optimal reconstruction might be hard
Proceedings of the twenty-sixth annual symposium on Computational geometry
Geodesic delaunay triangulations in bounded planar domains
ACM Transactions on Algorithms (TALG)
Proceedings of the twenty-seventh annual symposium on Computational geometry
Reeb graphs: approximation and persistence
Proceedings of the twenty-seventh 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
Multinerves and helly numbers of acyclic families
Proceedings of the twenty-eighth annual symposium on Computational geometry
Linear-size approximations to the vietoris-rips filtration
Proceedings of the twenty-eighth annual symposium on Computational geometry
Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes
Computational Geometry: Theory and Applications
Zigzag zoology: rips zigzags for homology inference
Proceedings of the twenty-ninth annual symposium on Computational geometry
Graph induced complex on point data
Proceedings of the twenty-ninth annual symposium on Computational geometry
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Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recent advances in higher dimensions have led to new methods to reconstruct large classes of compact subsets of Rd. However, the complexities of these methods scale up exponentially with d, making them impractical in medium or high dimensions, even on data sets of low intrinsic dimensionality. In this paper, we introduce a novel approach that stands in-between classical reconstruction and topological estimation, and whose complexity scales up with the intrinsic dimension of the data. Our algorithm combines two paradigms: greedy refinement, and topological persistence. Given a point cloud in Rd, we build a set of landmarks iteratively, while maintaining a nested pair of abstract complexes, whose images in Rd lie close to the data, and whose persistent homology eventually coincides with the homology of the underlying shape. When the data points are densely sampled from a smooth m-submanifold X of Rd, our method retrieves the homology of X in time at most c(m)n5, where n is the size of the input and c(m) is a constant depending solely on m. To prove the correctness of our algorithm, we investigate on Čech, Rips, and witness complex filtrations in Euclidean spaces. More precisely, we show how previous results on unions of balls can be transposed to Čech filtrations, and from there to Rips and witness complex filtrations. Finally, investigating further on witness complexes, we quantify a conjecture of Carlsson and de Silva, which states that witness complex filtrations should have cleaner persistence barcodes than Čech or Rips filtrations, at least on smooth submanifolds of Euclidean spaces.