A multi-resolution approach to heat kernels on discrete surfaces

Amir Vaxman; Mirela Ben-Chen; Craig Gotsman

doi:10.1145/1778765.1778858

A multi-resolution approach to heat kernels on discrete surfaces

Amir Vaxman
, Mirela Ben-Chen
, Craig Gotsman

Stanford University
Technion

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

Abstract

Studying the behavior of the heat diffusion process on a manifold is emerging as an important tool for analyzing the geometry of the manifold. Unfortunately, the high complexity of the computation of the heat kernel - the key to the diffusion process - limits this type of analysis to 3D models of modest resolution. We show how to use the unique properties of the heat kernel of a discrete two dimensional manifold to overcome these limitations. Combining a multi-resolution approach with a novel approximation method for the heat kernel at short times results in an efficient and robust algorithm for computing the heat kernels of detailed models. We show experimentally that our method can achieve good approximations in a fraction of the time required by traditional algorithms. Finally, we demonstrate how these heat kernels can be used to improve a diffusion-based feature extraction algorithm.

Original language	English
Title of host publication	ACM SIGGRAPH 2010 Papers, SIGGRAPH 2010
Editors	Hugues Hoppe
Publisher	Association for Computing Machinery
ISBN (Electronic)	9781450302104
DOIs	https://doi.org/10.1145/1778765.1778858
Publication status	Published - 26 Jul 2010
Event	37th International Conference and Exhibition on Computer Graphics and Interactive Techniques, SIGGRAPH 2010 - Los Angeles, United States Duration: 26 Jul 2010 → 30 Jul 2010

Publication series

Name	ACM SIGGRAPH 2010 Papers, SIGGRAPH 2010

Conference

Conference	37th International Conference and Exhibition on Computer Graphics and Interactive Techniques, SIGGRAPH 2010
Country/Territory	United States
City	Los Angeles
Period	26/07/10 → 30/07/10

Bibliographical note

Funding Information:
Thanks to Irad Yavneh for helpful numerical discussions. This work was partially supported by NSF grants 0808515 and 0914833, and by a joint Stanford-KAUST collaborative grant.

Publisher Copyright:
© 2010 ACM.

Funding

Thanks to Irad Yavneh for helpful numerical discussions. This work was partially supported by NSF grants 0808515 and 0914833, and by a joint Stanford-KAUST collaborative grant.

Keywords

Heat diffusion
Heat kernel
Matrix exponential
Multi-resolution

Access to Document

10.1145/1778765.1778858

Cite this

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title = "A multi-resolution approach to heat kernels on discrete surfaces",

abstract = "Studying the behavior of the heat diffusion process on a manifold is emerging as an important tool for analyzing the geometry of the manifold. Unfortunately, the high complexity of the computation of the heat kernel - the key to the diffusion process - limits this type of analysis to 3D models of modest resolution. We show how to use the unique properties of the heat kernel of a discrete two dimensional manifold to overcome these limitations. Combining a multi-resolution approach with a novel approximation method for the heat kernel at short times results in an efficient and robust algorithm for computing the heat kernels of detailed models. We show experimentally that our method can achieve good approximations in a fraction of the time required by traditional algorithms. Finally, we demonstrate how these heat kernels can be used to improve a diffusion-based feature extraction algorithm.",

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Vaxman, A, Ben-Chen, M & Gotsman, C 2010, A multi-resolution approach to heat kernels on discrete surfaces. in H Hoppe (ed.), ACM SIGGRAPH 2010 Papers, SIGGRAPH 2010., 121, ACM SIGGRAPH 2010 Papers, SIGGRAPH 2010, Association for Computing Machinery, 37th International Conference and Exhibition on Computer Graphics and Interactive Techniques, SIGGRAPH 2010, Los Angeles, United States, 26/07/10. https://doi.org/10.1145/1778765.1778858

A multi-resolution approach to heat kernels on discrete surfaces. / Vaxman, Amir; Ben-Chen, Mirela; Gotsman, Craig.
ACM SIGGRAPH 2010 Papers, SIGGRAPH 2010. ed. / Hugues Hoppe. Association for Computing Machinery, 2010. 121 (ACM SIGGRAPH 2010 Papers, SIGGRAPH 2010).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

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N1 - Funding Information: Thanks to Irad Yavneh for helpful numerical discussions. This work was partially supported by NSF grants 0808515 and 0914833, and by a joint Stanford-KAUST collaborative grant. Publisher Copyright: © 2010 ACM.

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N2 - Studying the behavior of the heat diffusion process on a manifold is emerging as an important tool for analyzing the geometry of the manifold. Unfortunately, the high complexity of the computation of the heat kernel - the key to the diffusion process - limits this type of analysis to 3D models of modest resolution. We show how to use the unique properties of the heat kernel of a discrete two dimensional manifold to overcome these limitations. Combining a multi-resolution approach with a novel approximation method for the heat kernel at short times results in an efficient and robust algorithm for computing the heat kernels of detailed models. We show experimentally that our method can achieve good approximations in a fraction of the time required by traditional algorithms. Finally, we demonstrate how these heat kernels can be used to improve a diffusion-based feature extraction algorithm.

AB - Studying the behavior of the heat diffusion process on a manifold is emerging as an important tool for analyzing the geometry of the manifold. Unfortunately, the high complexity of the computation of the heat kernel - the key to the diffusion process - limits this type of analysis to 3D models of modest resolution. We show how to use the unique properties of the heat kernel of a discrete two dimensional manifold to overcome these limitations. Combining a multi-resolution approach with a novel approximation method for the heat kernel at short times results in an efficient and robust algorithm for computing the heat kernels of detailed models. We show experimentally that our method can achieve good approximations in a fraction of the time required by traditional algorithms. Finally, we demonstrate how these heat kernels can be used to improve a diffusion-based feature extraction algorithm.

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A multi-resolution approach to heat kernels on discrete surfaces

Abstract

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Bibliographical note

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Keywords

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