7 Search Results for "Lieutier, André"

Document
DOI: 10.4230/LIPIcs.SoCG.2022.8
Delaunay-Like Triangulation of Smooth Orientable Submanifolds by 𝓁₁-Norm Minimization

Authors: Dominique Attali and André Lieutier

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
In this paper, we focus on one particular instance of the shape reconstruction problem, in which the shape we wish to reconstruct is an orientable smooth submanifold of the Euclidean space. Assuming we have as input a simplicial complex K that approximates the submanifold (such as the Čech complex or the Rips complex), we recast the reconstruction problem as a 𝓁₁-norm minimization problem in which the optimization variable is a chain of K. Providing that K satisfies certain reasonable conditions, we prove that the considered minimization problem has a unique solution which triangulates the submanifold and coincides with the flat Delaunay complex introduced and studied in a companion paper [D. Attali and A. Lieutier, 2022]. Since the objective is a weighted 𝓁₁-norm and the contraints are linear, the triangulation process can thus be implemented by linear programming.
Cite as

Dominique Attali and André Lieutier. Delaunay-Like Triangulation of Smooth Orientable Submanifolds by 𝓁₁-Norm Minimization. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{attali_et_al:LIPIcs.SoCG.2022.8,
  author =	{Attali, Dominique and Lieutier, Andr\'{e}},
  title =	{{Delaunay-Like Triangulation of Smooth Orientable Submanifolds by 𝓁₁-Norm Minimization}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{8:1--8:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.8},
  URN =		{urn:nbn:de:0030-drops-160162},
  doi =		{10.4230/LIPIcs.SoCG.2022.8},
  annote =	{Keywords: manifold reconstruction, Delaunay complex, triangulation, sampling conditions, optimization, 𝓁₁-norm minimization, simplicial complex, chain, fundamental class}
}
Document
Media Exposition
DOI: 10.4230/LIPIcs.SoCG.2022.66
A Cautionary Tale: Burning the Medial Axis Is Unstable (Media Exposition)

Authors: Erin Chambers, Christopher Fillmore, Elizabeth Stephenson, and Mathijs Wintraecken

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
The medial axis of a set consists of the points in the ambient space without a unique closest point on the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a topologically equivalent skeleton. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities various prunings of the medial axis have been proposed. Here, we examine one type of pruning, called burning. Because of the good experimental results, it was hoped that the burning method of simplifying the medial axis would be stable. In this work we show a simple example that dashes such hopes based on Bing’s house with two rooms, demonstrating an isotopy of a shape where the medial axis goes from collapsible to non-collapsible.
Cite as

Erin Chambers, Christopher Fillmore, Elizabeth Stephenson, and Mathijs Wintraecken. A Cautionary Tale: Burning the Medial Axis Is Unstable (Media Exposition). In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 66:1-66:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chambers_et_al:LIPIcs.SoCG.2022.66,
  author =	{Chambers, Erin and Fillmore, Christopher and Stephenson, Elizabeth and Wintraecken, Mathijs},
  title =	{{A Cautionary Tale: Burning the Medial Axis Is Unstable}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{66:1--66:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.66},
  URN =		{urn:nbn:de:0030-drops-160744},
  doi =		{10.4230/LIPIcs.SoCG.2022.66},
  annote =	{Keywords: Medial axis, Collapse, Pruning, Burning, Stability}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.20
The Topological Correctness of PL-Approximations of Isomanifolds

Authors: Jean-Daniel Boissonnat and Mathijs Wintraecken

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f: ℝ^d → ℝ^(d-n). A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation 𝒯. This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary.
Cite as

Jean-Daniel Boissonnat and Mathijs Wintraecken. The Topological Correctness of PL-Approximations of Isomanifolds. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{boissonnat_et_al:LIPIcs.SoCG.2020.20,
  author =	{Boissonnat, Jean-Daniel and Wintraecken, Mathijs},
  title =	{{The Topological Correctness of PL-Approximations of Isomanifolds}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{20:1--20:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.20},
  URN =		{urn:nbn:de:0030-drops-121787},
  doi =		{10.4230/LIPIcs.SoCG.2020.20},
  annote =	{Keywords: PL-approximations, isomanifolds, solution manifolds, topological correctness}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.32
Lexicographic Optimal Homologous Chains and Applications to Point Cloud Triangulations

Authors: David Cohen-Steiner, André Lieutier, and Julien Vuillamy

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
This paper considers a particular case of the Optimal Homologous Chain Problem (OHCP) for integer modulo 2 coefficients, where optimality is meant as a minimal lexicographic order on chains induced by a total order on simplices. The matrix reduction algorithm used for persistent homology is used to derive polynomial algorithms solving this problem instance, whereas OHCP is NP-hard in the general case. The complexity is further improved to a quasilinear algorithm by leveraging a dual graph minimum cut formulation when the simplicial complex is a pseudomanifold. We then show how this particular instance of the problem is relevant, by providing an application in the context of point cloud triangulation.
Cite as

David Cohen-Steiner, André Lieutier, and Julien Vuillamy. Lexicographic Optimal Homologous Chains and Applications to Point Cloud Triangulations. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cohensteiner_et_al:LIPIcs.SoCG.2020.32,
  author =	{Cohen-Steiner, David and Lieutier, Andr\'{e} and Vuillamy, Julien},
  title =	{{Lexicographic Optimal Homologous Chains and Applications to Point Cloud Triangulations}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{32:1--32:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.32},
  URN =		{urn:nbn:de:0030-drops-121908},
  doi =		{10.4230/LIPIcs.SoCG.2020.32},
  annote =	{Keywords: OHCP, simplicial homology, triangulation, Delaunay}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.54
Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex

Authors: Jisu Kim, Jaehyeok Shin, Frédéric Chazal, Alessandro Rinaldo, and Larry Wasserman

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
We derive conditions under which the reconstruction of a target space is topologically correct via the Čech complex or the Vietoris-Rips complex obtained from possibly noisy point cloud data. We provide two novel theoretical results. First, we describe sufficient conditions under which any non-empty intersection of finitely many Euclidean balls intersected with a positive reach set is contractible, so that the Nerve theorem applies for the restricted Čech complex. Second, we demonstrate the homotopy equivalence of a positive μ-reach set and its offsets. Applying these results to the restricted Čech complex and using the interleaving relations with the Čech complex (or the Vietoris-Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the Čech complex (or the Vietoris-Rips complex), in terms of the μ-reach. Our results sharpen existing results.
Cite as

Jisu Kim, Jaehyeok Shin, Frédéric Chazal, Alessandro Rinaldo, and Larry Wasserman. Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 54:1-54:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kim_et_al:LIPIcs.SoCG.2020.54,
  author =	{Kim, Jisu and Shin, Jaehyeok and Chazal, Fr\'{e}d\'{e}ric and Rinaldo, Alessandro and Wasserman, Larry},
  title =	{{Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{54:1--54:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.54},
  URN =		{urn:nbn:de:0030-drops-122129},
  doi =		{10.4230/LIPIcs.SoCG.2020.54},
  annote =	{Keywords: Computational topology, Homotopy reconstruction, Homotopy Equivalence, Vietoris-Rips complex, \v{C}ech complex, Reach, \mu-reach, Nerve Theorem, Offset, Double offset, Consistency}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.11
When Convexity Helps Collapsing Complexes

Authors: Dominique Attali, André Lieutier, and David Salinas

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
This paper illustrates how convexity hypotheses help collapsing simplicial complexes. We first consider a collection of compact convex sets and show that the nerve of the collection is collapsible whenever the union of sets in the collection is convex. We apply this result to prove that the Delaunay complex of a finite point set is collapsible. We then consider a convex domain defined as the convex hull of a finite point set. We show that if the point set samples sufficiently densely the domain, then both the Cech complex and the Rips complex of the point set are collapsible for a well-chosen scale parameter. A key ingredient in our proofs consists in building a filtration by sweeping space with a growing sphere whose center has been fixed and studying events occurring through the filtration. Since the filtration mimics the sublevel sets of a Morse function with a single critical point, we anticipate this work to lay the foundations for a non-smooth, discrete Morse Theory.
Cite as

Dominique Attali, André Lieutier, and David Salinas. When Convexity Helps Collapsing Complexes. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{attali_et_al:LIPIcs.SoCG.2019.11,
  author =	{Attali, Dominique and Lieutier, Andr\'{e} and Salinas, David},
  title =	{{When Convexity Helps Collapsing Complexes}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.11},
  URN =		{urn:nbn:de:0030-drops-104152},
  doi =		{10.4230/LIPIcs.SoCG.2019.11},
  annote =	{Keywords: collapsibility, convexity, collection of compact convex sets, nerve, filtration, Delaunay complex, Cech complex, Rips complex}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.10
The Reach, Metric Distortion, Geodesic Convexity and the Variation of Tangent Spaces

Authors: Jean-Daniel Boissonnat, André Lieutier, and Mathijs Wintraecken

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
In this paper we discuss three results. The first two concern general sets of positive reach: We first characterize the reach by means of a bound on the metric distortion between the distance in the ambient Euclidean space and the set of positive reach. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the distance between the points and the reach.
Cite as

Jean-Daniel Boissonnat, André Lieutier, and Mathijs Wintraecken. The Reach, Metric Distortion, Geodesic Convexity and the Variation of Tangent Spaces. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{boissonnat_et_al:LIPIcs.SoCG.2018.10,
  author =	{Boissonnat, Jean-Daniel and Lieutier, Andr\'{e} and Wintraecken, Mathijs},
  title =	{{The Reach, Metric Distortion, Geodesic Convexity and the Variation of Tangent Spaces}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{10:1--10:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.10},
  URN =		{urn:nbn:de:0030-drops-87236},
  doi =		{10.4230/LIPIcs.SoCG.2018.10},
  annote =	{Keywords: Reach, Metric distortion, Manifolds, Convexity}
}
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