research-article

Spectral coarsening of geometric operators

Authors:

Hsueh-Ti Derek Liu,

Maks OvsjanikovAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 38, Issue 4

Article No.: 105, Pages 1 - 13

https://doi.org/10.1145/3306346.3322953

Published: 12 July 2019 Publication History

Abstract

We introduce a novel approach to measure the behavior of a geometric operator before and after coarsening. By comparing eigenvectors of the input operator and its coarsened counterpart, we can quantitatively and visually analyze how well the spectral properties of the operator are maintained. Using this measure, we show that standard mesh simplification and algebraic coarsening techniques fail to maintain spectral properties. In response, we introduce a novel approach for spectral coarsening. We show that it is possible to significantly reduce the sampling density of an operator derived from a 3D shape without affecting the low-frequency eigenvectors. By marrying techniques developed within the algebraic multigrid and the functional maps literatures, we successfully coarsen a variety of isotropic and anisotropic operators while maintaining sparsity and positive semi-definiteness. We demonstrate the utility of this approach for applications including operatorsensitive sampling, shape matching, and graph pooling for convolutional neural networks.

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Maggioli FBaieri DRodolà EMelzi S(2024)ReMatching: Low-Resolution Representations for Scalable Shape CorrespondenceComputer Vision – ECCV 202410.1007/978-3-031-72913-3_11(183-200)Online publication date: 2-Dec-2024
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Index Terms

Spectral coarsening of geometric operators
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Shape analysis
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on matrices

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cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 38, Issue 4

August 2019

1480 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3306346

Editor:
Olga Sorkine-Hornung
ETH Zurich

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Copyright © 2019 ACM.

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Publication History

Published: 12 July 2019

Published in TOG Volume 38, Issue 4

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Canada Excellence Research Chairs, Government of Canada
European Research Council
MESH
Natural Sciences and Engineering Research Council of Canada
Fields Institute CQAM Labs
Mitacs Globalink
Autodesk
Adobe Inc.

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Cited By

Yazgan MSahillioğlu Y(2024)A Partition Based Method for Spectrum-Preserving Mesh SimplificationIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2023.334161030:10(6839-6850)Online publication date: Oct-2024
https://doi.org/10.1109/TVCG.2023.3341610
Maggioli FBaieri DRodolà EMelzi S(2024)ReMatching: Low-Resolution Representations for Scalable Shape CorrespondenceComputer Vision – ECCV 202410.1007/978-3-031-72913-3_11(183-200)Online publication date: 2-Dec-2024
https://doi.org/10.1007/978-3-031-72913-3_11
Montes Maestre JDu YHinchet RCoros SThomaszewski B(2023)Differentiable Stripe Patterns for Inverse Design of Structured SurfacesACM Transactions on Graphics10.1145/359211442:4(1-14)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592114
Keros ASubr K(2023)Spectral Coarsening with Hodge LaplaciansACM SIGGRAPH 2023 Conference Proceedings10.1145/3588432.3591544(1-11)Online publication date: 23-Jul-2023
https://dl.acm.org/doi/10.1145/3588432.3591544
Magnet ROvsjanikov M(2023)Scalable and Efficient Functional Map Computations on Dense MeshesComputer Graphics Forum10.1111/cgf.1474642:2(89-101)Online publication date: 23-May-2023
https://doi.org/10.1111/cgf.14746
Farazi MYang ZZhu WQiu PWang Y(2023)TetCNN: Convolutional Neural Networks on Tetrahedral MeshesInformation Processing in Medical Imaging10.1007/978-3-031-34048-2_24(303-315)Online publication date: 8-Jun-2023
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Duenser SThomaszewski BPoranne RCoros S(2022)Nonlinear Compliant Modes for Large-deformation Analysis of Flexible StructuresACM Transactions on Graphics10.1145/356895242:2(1-11)Online publication date: 22-Nov-2022
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Neveu WPuhachov IThomaszewski BBessmeltsev M(2022)Stability-Aware Simplification of Curve NetworksACM SIGGRAPH 2022 Conference Proceedings10.1145/3528233.3530711(1-9)Online publication date: 27-Jul-2022
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Nasikun AHildebrandt K(2022)The Hierarchical Subspace Iteration Method for Laplace–Beltrami EigenproblemsACM Transactions on Graphics10.1145/349520841:2(1-14)Online publication date: 4-Jan-2022
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Deng BYao YDyke RZhang J(2022)A Survey of Non‐Rigid 3D RegistrationComputer Graphics Forum10.1111/cgf.1450241:2(559-589)Online publication date: 24-May-2022
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