2 Search Results for "Kachanovich, Siargey"

Document
DOI: 10.4230/LIPIcs.SoCG.2021.17
Tracing Isomanifolds in ℝ^d in Time Polynomial in d Using Coxeter-Freudenthal-Kuhn Triangulations

Authors: Jean-Daniel Boissonnat, Siargey Kachanovich, and Mathijs Wintraecken

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. submanifolds of ℝ^d defined as the zero set of some multivariate multivalued smooth function f: ℝ^d → ℝ^{d-n}, where n is the intrinsic dimension of the manifold. A natural way to approximate a smooth isomanifold M is to consider its Piecewise-Linear (PL) approximation M̂ based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we describe a simple algorithm to trace isomanifolds from a given starting point. The algorithm works for arbitrary dimensions n and d, and any precision D. Our main result is that, when f (or M) has bounded complexity, the complexity of the algorithm is polynomial in d and δ = 1/D (and unavoidably exponential in n). Since it is known that for δ = Ω (d^{2.5}), M̂ is O(D²)-close and isotopic to M, our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality reduction techniques, the dependency on d in the size of M̂ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art.
Cite as

Jean-Daniel Boissonnat, Siargey Kachanovich, and Mathijs Wintraecken. Tracing Isomanifolds in ℝ^d in Time Polynomial in d Using Coxeter-Freudenthal-Kuhn Triangulations. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{boissonnat_et_al:LIPIcs.SoCG.2021.17,
  author =	{Boissonnat, Jean-Daniel and Kachanovich, Siargey and Wintraecken, Mathijs},
  title =	{{Tracing Isomanifolds in \mathbb{R}^d in Time Polynomial in d Using Coxeter-Freudenthal-Kuhn Triangulations}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{17:1--17:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.17},
  URN =		{urn:nbn:de:0030-drops-138163},
  doi =		{10.4230/LIPIcs.SoCG.2021.17},
  annote =	{Keywords: Coxeter triangulation, Kuhn triangulation, permutahedron, PL-approximations, isomanifolds/solution manifolds/isosurfacing}
}
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)

Copy BibTex To Clipboard

@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}
}
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