The crust and the &Bgr;-Skeleton: combinatorial curve reconstruction
Graphical Models and Image Processing
Delaunay triangulations and Voronoi diagrams for Riemannian manifolds
Proceedings of the sixteenth annual symposium on Computational geometry
Discrete & Computational Geometry
Manifold reconstruction from point samples
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Provably good sampling and meshing of Lipschitz surfaces
Proceedings of the twenty-second annual symposium on Computational geometry
Boundary recognition in sensor networks by topological methods
Proceedings of the 12th annual international conference on Mobile computing and networking
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
SGP '07 Proceedings of the fifth Eurographics symposium on Geometry processing
Reconstruction using witness complexes
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
A Discrete Laplace–Beltrami Operator for Simplicial Surfaces
Discrete & Computational Geometry
Geodesic Delaunay triangulation and witness complex in the plane
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
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
Analysis of scalar fields over point cloud data
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Proximity of persistence modules and their diagrams
Proceedings of the twenty-fifth annual symposium on Computational geometry
Provable surface reconstruction from noisy samples
Computational Geometry: Theory and Applications
Topological estimation using witness complexes
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
Graph induced complex on point data
Proceedings of the twenty-ninth annual symposium on Computational geometry
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We introduce a new feature size for bounded domains in the plane endowed with an intrinsic metric. Given a point x in a domain X, the systolic feature size of X at x measures half the length of the shortest loop through x that is not null-homotopic in X. The resort to an intrinsic metric makes the systolic feature size rather insensitive to the local geometry of the domain, in contrast with its predecessors (local feature size, weak feature size, homology feature size). This reduces the number of samples required to capture the topology of X, provided that a reliable approximation to the intrinsic metric of X is available. Under sufficient sampling conditions involving the systolic feature size, we show that the geodesic Delaunay triangulation Dx(L) of a finite sampling L is homotopy equivalent to X. Under similar conditions, Dx(L) is sandwiched between the geodesic witness complex CWX(L) and a relaxed version CWX,ν(L). In the conference version of the article, we took advantage of this fact and proved that the homology of Dx(L) (and hence the one of X) can be retrieved by computing the persistent homology between CWX(L) and CWX,ν(L). Here, we investigate further and show that the homology of X can also be recovered from the persistent homology associated with inclusions of type CWX,ν(L)↪CWX,ν′(L), under some conditions on the parameters ν≤ν′. Similar results are obtained for Vietoris-Rips complexes in the intrinsic metric. The proofs draw some connections with recent advances on the front of homology inference from point cloud data, but also with several well-known concepts of Riemannian (and even metric) geometry. On the algorithmic front, we propose algorithms for estimating the systolic feature size of a bounded planar domain X, selecting a landmark set of sufficient density, and computing the homology of X using geodesic witness complexes or Rips complexes.