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Computing a high-dimensional euclidean embedding from an arbitrary smooth riemannian metric

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Xiaohu GuoAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 37, Issue 4

Article No.: 62, Pages 1 - 16

https://doi.org/10.1145/3197517.3201369

Published: 30 July 2018 Publication History

Abstract

This article presents a new method to compute a self-intersection free high-dimensional Euclidean embedding (SIFHDE²) for surfaces and volumes equipped with an arbitrary Riemannian metric. It is already known that given a high-dimensional (high-d) embedding, one can easily compute an anisotropic Voronoi diagram by back-mapping it to 3D space. We show here how to solve the inverse problem, i.e., given an input metric, compute a smooth intersection-free high-d embedding of the input such that the pullback metric of the embedding matches the input metric. Our numerical solution mechanism matches the deformation gradient of the 3D → higher-d mapping with the given Riemannian metric. We demonstrate the applicability of our method, by using it to construct anisotropic Restricted Voronoi Diagram (RVD) and anisotropic meshing, that are otherwise extremely difficult to compute. In SIFHDE²-space constructed by our algorithm, difficult 3D anisotropic computations are replaced with simple Euclidean computations, resulting in an isotropic RVD and its dual mesh on this high-d embedding. Results are compared with the state-of-the-art in anisotropic surface and volume meshings using several examples and evaluation metrics.

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Dai YSu JFu X(2024)Anisotropic triangular meshing using metric-adapted embeddingsComputer Aided Geometric Design10.1016/j.cagd.2024.102314111(102314)Online publication date: Jun-2024
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Index Terms

Computing a high-dimensional euclidean embedding from an arbitrary smooth riemannian metric
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Parametric curve and surface models
      2. Volumetric models

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cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 37, Issue 4

August 2018

1670 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3197517

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Copyright © 2018 ACM.

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Published: 30 July 2018

Published in TOG Volume 37, Issue 4

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Nielsen F(2024)Approximation and bounding techniques for the Fisher-Rao distances between parametric statistical modelsProbability Models10.1016/bs.host.2024.06.003(67-116)Online publication date: 2024
https://doi.org/10.1016/bs.host.2024.06.003
Louis-Rosenberg J(2023)Building a Gyroid from Flat SheetsProceedings of the 8th ACM Symposium on Computational Fabrication10.1145/3623263.3629158(1-2)Online publication date: 8-Oct-2023
https://dl.acm.org/doi/10.1145/3623263.3629158
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Xia QZhang JFang ZLi JZhang MDeng BHe Y(2022)GeodesicEmbedding (GE): A High-Dimensional Embedding Approach for Fast Geodesic Distance QueriesIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2021.310997528:12(4930-4939)Online publication date: 1-Dec-2022
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