Isospectralization, or how to hear shape, style, and correspondence

Luca Cosmo; Mikhail Panine; Arianna Rampini; Maks Ovsjanikov; Michael M. Bronstein; Emanuele Rodola

doi:10.1109/CVPR.2019.00771

Isospectralization, or how to hear shape, style, and correspondence

Luca Cosmo, Mikhail Panine, Arianna Rampini, Maks Ovsjanikov, Michael M. Bronstein, Emanuele Rodola

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

43 Scopus citations

Abstract

The question whether one can recover the shape of a geometric object from its Laplacian spectrum ('hear the shape of the drum') is a classical problem in spectral geometry with a broad range of implications and applications. While theoretically the answer to this question is negative (there exist examples of iso-spectral but non-isometric manifolds), little is known about the practical possibility of using the spectrum for shape reconstruction and optimization. In this paper, we introduce a numerical procedure called isospectralization, consisting of deforming one shape to make its Laplacian spectrum match that of another. We implement the isospectralization procedure using modern differentiable programming techniques and exemplify its applications in some of the classical and notoriously hard problems in geometry processing, computer vision, and graphics such as shape reconstruction, pose and style transfer, and dense deformable correspondence.

Original language	English (US)
Title of host publication	2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)
Publisher	IEEE
Pages	7521-7530
Number of pages	10
ISBN (Print)	9781728132938
DOIs	https://doi.org/10.1109/CVPR.2019.00771
State	Published - Jan 9 2020
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2022-06-24
Acknowledged KAUST grant number(s): OSR-CRG2017-3426
Acknowledgements: The authors wish to thank Alex Bronstein for useful discussions. ER and AR are supported by the ERC Starting Grant No. 802554 (SPECGEO). MB and LC are partially supported by ERC Consolidator Grant No. 724228 (LEMAN) and Google Research Faculty awards. MB is also partially supported by the Royal Society Wolfson Research Merit award and Rudolf Diesel industrial fellowship at TU Munich. Parts of this work were supported by a Google Focused Research Award, KAUST OSR Award No. OSR-CRG2017-3426, a gift from the NVIDIA Corporation and the ERC Starting Grant No. 758800 (EXPROTEA).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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10.1109/CVPR.2019.00771

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title = "Isospectralization, or how to hear shape, style, and correspondence",

abstract = "The question whether one can recover the shape of a geometric object from its Laplacian spectrum ('hear the shape of the drum') is a classical problem in spectral geometry with a broad range of implications and applications. While theoretically the answer to this question is negative (there exist examples of iso-spectral but non-isometric manifolds), little is known about the practical possibility of using the spectrum for shape reconstruction and optimization. In this paper, we introduce a numerical procedure called isospectralization, consisting of deforming one shape to make its Laplacian spectrum match that of another. We implement the isospectralization procedure using modern differentiable programming techniques and exemplify its applications in some of the classical and notoriously hard problems in geometry processing, computer vision, and graphics such as shape reconstruction, pose and style transfer, and dense deformable correspondence.",

author = "Luca Cosmo and Mikhail Panine and Arianna Rampini and Maks Ovsjanikov and Bronstein, {Michael M.} and Emanuele Rodola",

note = "KAUST Repository Item: Exported on 2022-06-24 Acknowledged KAUST grant number(s): OSR-CRG2017-3426 Acknowledgements: The authors wish to thank Alex Bronstein for useful discussions. ER and AR are supported by the ERC Starting Grant No. 802554 (SPECGEO). MB and LC are partially supported by ERC Consolidator Grant No. 724228 (LEMAN) and Google Research Faculty awards. MB is also partially supported by the Royal Society Wolfson Research Merit award and Rudolf Diesel industrial fellowship at TU Munich. Parts of this work were supported by a Google Focused Research Award, KAUST OSR Award No. OSR-CRG2017-3426, a gift from the NVIDIA Corporation and the ERC Starting Grant No. 758800 (EXPROTEA). This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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doi = "10.1109/CVPR.2019.00771",

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AU - Cosmo, Luca

AU - Panine, Mikhail

AU - Rampini, Arianna

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AU - Bronstein, Michael M.

AU - Rodola, Emanuele

N1 - KAUST Repository Item: Exported on 2022-06-24 Acknowledged KAUST grant number(s): OSR-CRG2017-3426 Acknowledgements: The authors wish to thank Alex Bronstein for useful discussions. ER and AR are supported by the ERC Starting Grant No. 802554 (SPECGEO). MB and LC are partially supported by ERC Consolidator Grant No. 724228 (LEMAN) and Google Research Faculty awards. MB is also partially supported by the Royal Society Wolfson Research Merit award and Rudolf Diesel industrial fellowship at TU Munich. Parts of this work were supported by a Google Focused Research Award, KAUST OSR Award No. OSR-CRG2017-3426, a gift from the NVIDIA Corporation and the ERC Starting Grant No. 758800 (EXPROTEA). This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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N2 - The question whether one can recover the shape of a geometric object from its Laplacian spectrum ('hear the shape of the drum') is a classical problem in spectral geometry with a broad range of implications and applications. While theoretically the answer to this question is negative (there exist examples of iso-spectral but non-isometric manifolds), little is known about the practical possibility of using the spectrum for shape reconstruction and optimization. In this paper, we introduce a numerical procedure called isospectralization, consisting of deforming one shape to make its Laplacian spectrum match that of another. We implement the isospectralization procedure using modern differentiable programming techniques and exemplify its applications in some of the classical and notoriously hard problems in geometry processing, computer vision, and graphics such as shape reconstruction, pose and style transfer, and dense deformable correspondence.

AB - The question whether one can recover the shape of a geometric object from its Laplacian spectrum ('hear the shape of the drum') is a classical problem in spectral geometry with a broad range of implications and applications. While theoretically the answer to this question is negative (there exist examples of iso-spectral but non-isometric manifolds), little is known about the practical possibility of using the spectrum for shape reconstruction and optimization. In this paper, we introduce a numerical procedure called isospectralization, consisting of deforming one shape to make its Laplacian spectrum match that of another. We implement the isospectralization procedure using modern differentiable programming techniques and exemplify its applications in some of the classical and notoriously hard problems in geometry processing, computer vision, and graphics such as shape reconstruction, pose and style transfer, and dense deformable correspondence.

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BT - 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)

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Isospectralization, or how to hear shape, style, and correspondence

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