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Functional Characterization of Intrinsic and Extrinsic Geometry

Authors:

Etienne Corman,

Justin Solomon,

Mirela Ben-Chen,

Leonidas Guibas,

Maks OvsjanikovAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 36, Issue 4

Article No.: 59a, Page 1

https://doi.org/10.1145/3072959.2999535

Published: 16 July 2017 Publication History

Abstract

We propose a novel way to capture and characterize distortion between pairs of shapes by extending the recently proposed framework of shape differences built on functional maps. We modify the original definition of shape differences slightly and prove that after this change, the discrete metric is fully encoded in two shape difference operators and can be recovered by solving two linear systems of equations. Then we introduce an extension of the shape difference operators using offset surfaces to capture extrinsic or embedding-dependent distortion, complementing the purely intrinsic nature of the original shape differences. Finally, we demonstrate that a set of four operators is complete, capturing intrinsic and extrinsic structure and fully encoding a shape up to rigid motion in both discrete and continuous settings. We highlight the usefulness of our constructions by showing the complementary nature of our extrinsic shape differences in capturing distortion ignored by previous approaches. We additionally provide examples where we recover local shape structure from the shape difference operators, suggesting shape editing and analysis tools based on manipulating shape differences.

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Cited By

Hartwig FSassen JAzencot ORumpf MBen-Chen M(2023)An Elastic Basis for Spectral Shape CorrespondenceACM SIGGRAPH 2023 Conference Proceedings10.1145/3588432.3591518(1-11)Online publication date: 23-Jul-2023
https://dl.acm.org/doi/10.1145/3588432.3591518
Smirnov DSolomon J(2021)HodgeNetACM Transactions on Graphics10.1145/3450626.345979740:4(1-11)Online publication date: Aug-2021
https://doi.org/10.1145/3450626.3459797
Mo KGuerrero PYi LSu HWonka PMitra NGuibas L(2020)StructEdit: Learning Structural Shape Variations2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)10.1109/CVPR42600.2020.00888(8856-8865)Online publication date: Jun-2020
https://doi.org/10.1109/CVPR42600.2020.00888
Show More Cited By

Index Terms

Functional Characterization of Intrinsic and Extrinsic Geometry
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Shape analysis

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Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 36, Issue 4

August 2017

2155 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3072959

Issue’s Table of Contents

Copyright © 2017 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 16 July 2017

Accepted: 01 December 2016

Revised: 01 October 2016

Received: 01 October 2015

Published in TOG Volume 36, Issue 4

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Google Focused Research Award
Marie Curie CIG
CNRS chaire d'excellence, chaire Jean Marjoulet from École Polytechnique, FUI project TANDEM 2
Direction Générale de l’Armement (DGA)
NSF Mathematical Sciences Postdoctoral Research Fellowship
NSF
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Cited By

Hartwig FSassen JAzencot ORumpf MBen-Chen M(2023)An Elastic Basis for Spectral Shape CorrespondenceACM SIGGRAPH 2023 Conference Proceedings10.1145/3588432.3591518(1-11)Online publication date: 23-Jul-2023
https://dl.acm.org/doi/10.1145/3588432.3591518
Smirnov DSolomon J(2021)HodgeNetACM Transactions on Graphics10.1145/3450626.345979740:4(1-11)Online publication date: Aug-2021
https://doi.org/10.1145/3450626.3459797
Mo KGuerrero PYi LSu HWonka PMitra NGuibas L(2020)StructEdit: Learning Structural Shape Variations2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)10.1109/CVPR42600.2020.00888(8856-8865)Online publication date: Jun-2020
https://doi.org/10.1109/CVPR42600.2020.00888
Sassen JHeeren BHildebrandt KRumpf M(2020)Geometric optimization using nonlinear rotation-invariant coordinatesComputer Aided Geometric Design10.1016/j.cagd.2020.10182977(101829)Online publication date: Feb-2020
https://doi.org/10.1016/j.cagd.2020.101829
Aumentado-Armstrong TTsogkas SJepson ADickinson S(2019)Geometric Disentanglement for Generative Latent Shape Models2019 IEEE/CVF International Conference on Computer Vision (ICCV)10.1109/ICCV.2019.00827(8180-8189)Online publication date: Oct-2019
https://doi.org/10.1109/ICCV.2019.00827
Ovsjanikov MCorman EBronstein MRodolà EBen-Chen MGuibas LChazal FBronstein AMitra N(2016)Computing and processing correspondences with functional mapsSIGGRAPH ASIA 2016 Courses10.1145/2988458.2988494(1-60)Online publication date: 28-Nov-2016
https://dl.acm.org/doi/10.1145/2988458.2988494

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