The union of balls and its dual shape
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
A Delaunay refinement algorithm for quality 2-dimensional mesh generation
SODA '93 Selected papers from the fourth annual ACM SIAM symposium on Discrete algorithms
Nice point sets can have nasty Delaunay triangulations
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Discrete & Computational Geometry
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Stability and Computation of Topological Invariants of Solids in ${\Bbb R}^n$
Discrete & Computational Geometry
Finding the Homology of Submanifolds with High Confidence from Random Samples
Discrete & Computational Geometry
Towards persistence-based reconstruction in euclidean spaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Reconstruction Using Witness Complexes
Discrete & Computational Geometry
Computational Geometry: Theory and Applications
Size complexity of volume meshes vs. surface meshes
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Extending Persistence Using Poincaré and Lefschetz Duality
Foundations of Computational Mathematics
Proximity of persistence modules and their diagrams
Proceedings of the twenty-fifth annual symposium on Computational geometry
Topological estimation using witness complexes
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
An output-sensitive algorithm for persistent homology
Proceedings of the twenty-seventh annual symposium on Computational geometry
Beating the spread: time-optimal point meshing
Proceedings of the twenty-seventh annual symposium on Computational geometry
Linear-size approximations to the vietoris-rips filtration
Proceedings of the twenty-eighth annual symposium on Computational geometry
New Bounds on the Size of Optimal Meshes
Computer Graphics Forum
An output-sensitive algorithm for persistent homology
Computational Geometry: Theory and Applications
Efficient computation of clipped Voronoi diagram for mesh generation
Computer-Aided Design
Zigzag zoology: rips zigzags for homology inference
Proceedings of the twenty-ninth annual symposium on Computational geometry
A fast algorithm for well-spaced points and approximate delaunay graphs
Proceedings of the twenty-ninth annual symposium on Computational geometry
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We apply ideas from mesh generation to improve the time and space complexities of computing the full persistent homological information associated with a point cloud P in Euclidean space ℜd. Classical approaches rely on the Cech, Rips, ±-complex, or witness complex filtrations of P, whose complexities scale up very badly with d. For instance, the ±-complex filtration incurs the n Ω(d) size of the Delaunay triangulation, where n is the size of P. The common alternative is to truncate the filtrations when the sizes of the complexes become prohibitive, possibly before discovering the most relevant topological features. In this paper we propose a new collection of filtrations, based on the Delaunay triangulation of a carefully-chosen superset of P, whose sizes are reduced to 2O(d2)n. Our filtrations interleave multiplicatively with the family of offsets of P, so that the persistence diagram of P can be approximated in 2O(d2)n3 time in theory, with a near-linear observed running time in practice. Thus, our approach remains tractable in medium dimensions, say 4 to 10.