A Sampling Theory for Compact Sets in Euclidean Space

Chazal, Frédéric; Cohen-Steiner, David; Lieutier, André

doi:10.1007/s00454-009-9144-8

Cited by 134 publications

(178 citation statements)

References 33 publications

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“…A stability result of µ-critical points when the ambient space is Euclidean is proved in [7]. We prove a generalization of this result for when the curvature of the ambient space is bounded from below.…”

Section: Introductionmentioning

confidence: 90%

“…This can be thought of as the obvious generalization of the gradient vector fields for distance functions from compact subsets of Euclidean space (as studied in [7] and [17]). …”

Section: Stability Of µ-Critical Pointsmentioning

confidence: 99%

“…The following proposition is a generalization of critical point stability theorem in [7] where the ambient space can now be any manifold with non-negative curvature.…”

Section: Y)} Then X Is a Critical Point If And Only If 0 Lies In Thementioning

confidence: 99%

“…To deal with a larger class of sets Chazal, Cohen-Steiner and Lieutier in [7] introduced the notion of µ-reach. A point is µ-critical when the norm of the gradient of the distance function at that point is less than or equal to µ.…”

Section: Introductionmentioning

confidence: 99%

“…The µ-reach of a set is the supremum of r > 0 such that the r offset does not contain any µ-critical points. In [7] and [2] sampling conditions are given in terms of the µ-reach.…”

Section: Introductionmentioning

confidence: 99%

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Cone Fields and Topological Sampling in Manifolds with Bounded Curvature

Turner

2013

Found Comput Math

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A standard reconstruction problem is how to discover a compact set from a noisy point cloud that approximates it. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the Hausdorff distance between two compact sets, for when certain offsets of these two sets are homotopic in terms of the absence of µ-critical points in an annular region. We reduce the problem of reconstructing a subset from a point cloud to the existence of a deformation retraction from the offset of the subset to the subset itself. The ambient space can be any Riemannian manifold but we focus on ambient manifolds which have nowhere negative curvature (this includes Euclidean space). We get an improvement on previous bounds for the case where the ambient space is Euclidean whenever µ ≤ 0.945 (µ ∈ (0, 1) by definition). In the process, we prove stability theorems for µ-critical points when the ambient space is a manifold.

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Section: Introductionmentioning

confidence: 90%

“…This can be thought of as the obvious generalization of the gradient vector fields for distance functions from compact subsets of Euclidean space (as studied in [7] and [17]). …”

Section: Stability Of µ-Critical Pointsmentioning

confidence: 99%

“…The following proposition is a generalization of critical point stability theorem in [7] where the ambient space can now be any manifold with non-negative curvature.…”

Section: Y)} Then X Is a Critical Point If And Only If 0 Lies In Thementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The µ-reach of a set is the supremum of r > 0 such that the r offset does not contain any µ-critical points. In [7] and [2] sampling conditions are given in terms of the µ-reach.…”

Section: Introductionmentioning

confidence: 99%

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Cone Fields and Topological Sampling in Manifolds with Bounded Curvature

Turner

2013

Found Comput Math

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On the vanishing of homology in random Čech complexes

Bobrowski

Weinberger²

2016

Random Struct. Alg.

120

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We compute the homology of randomČech complexes over a homogeneous Poisson process on the d-dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to the Erdős -Rényi phase transition, where theČech complex becomes connected. The second transition is where all the other homology groups are computed correctly (almost simultaneously). Our calculations also suggest a finer measurement of scales, where there is a further refinement to this picture and separation between different homology groups.

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Random Čech complexes on Riemannian manifolds

Bobrowski

Oliveira

2018

Random Struct Algorithms

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In this paper we study the homology of a random Čech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for “homological connectivity” where the homology of the complex becomes isomorphic to that of M. The results presented in this paper are an important generalization of , from the flat torus to general compact Riemannian manifolds. In addition to proving the statements related to homological connectivity, the methods we develop in this paper can be used as a framework for translating results for random geometric graphs and complexes from the Euclidean setting into the more general Riemannian one.

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A Sampling Theory for Compact Sets in Euclidean Space

Cited by 134 publications

References 33 publications

Cone Fields and Topological Sampling in Manifolds with Bounded Curvature

Cone Fields and Topological Sampling in Manifolds with Bounded Curvature

On the vanishing of homology in random Čech complexes

Random Čech complexes on Riemannian manifolds

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