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
Persistence barcodes for shapes
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
SCG '05 Proceedings of the twenty-first annual symposium on 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
Greedy optimal homotopy and homology generators
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
A sampling theory for compact sets in Euclidean space
Proceedings of the twenty-second annual symposium on Computational geometry
Provably good sampling and meshing of Lipschitz surfaces
Proceedings of the twenty-second annual symposium on Computational geometry
Coverage and hole-detection in sensor networks via homology
IPSN '05 Proceedings of the 4th international symposium on Information processing in sensor networks
Provable surface reconstruction from noisy samples
Computational Geometry: Theory and Applications
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
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
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
Connectivity-based localization of large-scale sensor networks with complex shape
ACM Transactions on Sensor Networks (TOSN)
Surface sampling and the intrinsic Voronoi diagram
SGP '08 Proceedings of the Symposium on Geometry Processing
Geodesic delaunay triangulations in bounded planar domains
ACM Transactions on Algorithms (TALG)
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We introduce a novel feature size for bounded planar domains endowed with an intrinsic metric. Given a point x in such a domain X, the homotopy feature size of X at x, or hfs(x) for short, measures half the length of the shortest loop through x that is not null-homotopic in X. The resort to an intrinsic metric makes hfs(x) rather insensitive to the local geometry of X, in contrast with its predecessors (local feature size, weak feature size, homology feature size). This leads to a reduced number of samples that still capture the topology of X. Under reasonable sampling conditions involving hfs, we show that the geodesic Delaunay traingulation DX (L) of a finite sampling L of X is homotopy equivalent to X. Moreover, DX (L) is sandwiched between the geodesic witness complex CWX (L) and a relaxed version CWX, v (L), defined by a parameter v. Taking advantage of this fact, we prove that the homology of DX (L) (and hence of X) can be retrieved by computing the persistent homology between CWX (L) and CWX, v (L). We propose algorithms for estimating hfs, selecting a landmark set of sufficient density, building its geodesic Delaunay triangulation, and computing the homology of X using CWX (L). We also present some simulation results in the context of sensor networks that corroborate our theoretical statements.