Learning Metric Fields for Fast Low‐Distortion Mesh Parameterizations.
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| Title: | Learning Metric Fields for Fast Low‐Distortion Mesh Parameterizations. |
|---|---|
| Authors: | Fargion, G.1 (AUTHOR), Weber, O.1 (AUTHOR) |
| Source: | Computer Graphics Forum. May2025, Vol. 44 Issue 2, p1-14. 14p. |
| Subjects: | Harmonic maps, Tensor fields, Artificial neural networks, Laplacian matrices, Metric spaces, Injective functions |
| Abstract: | We present a fast and robust method for computing an injective parameterization with low isometric distortion for disk‐like triangular meshes. Harmonic function‐based methods, with their rich mathematical foundation, are widely used. Harmonic maps are particularly valuable for ensuring injectivity under certain boundary conditions. In addition, they offer computational efficiency by forming a linear subspace [FW22]. However, this restricted subspace often leads to significant isometric distortion, especially for highly curved surfaces. Conversely, methods that operate in the full space of piecewise linear maps [SPSH*17] achieve lower isometric distortion, but at a higher computational cost. Aigerman et al. [AGK*22] pioneered a parameterization method that uses deep neural networks to predict the Jacobians of the map at mesh triangles, and integrates them into an explicit map by solving a Poisson equation. However, this approach often results in significant Poisson reconstruction errors due to the inability to ensure the integrability of the predicted neural Jacobian field, leading to unbounded distortion and lack of local injectivity. We propose a hybrid method that combines the speed and robustness of harmonic maps with the generality of deep neural networks to produce injective maps with low isometric distortion much faster than state‐of‐the‐art methods. The core concept is simple but powerful. Instead of learning Jacobian fields, we learn metric tensor fields over the input mesh, resulting in a cus‐tomized Laplacian matrix that defines a harmonic map in a modified metric [WGS23]. Our approach ensures injectivity, offers great computational efficiency, and produces significantly lower isometric distortion compared to straightforward harmonic maps. [ABSTRACT FROM AUTHOR] |
| Copyright of Computer Graphics Forum is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 186773121 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Learning Metric Fields for Fast Low‐Distortion Mesh Parameterizations. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Fargion%2C+G%2E%22">Fargion, G.</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Weber%2C+O%2E%22">Weber, O.</searchLink><relatesTo>1</relatesTo> (AUTHOR) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Computer+Graphics+Forum%22">Computer Graphics Forum</searchLink>. May2025, Vol. 44 Issue 2, p1-14. 14p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Harmonic+maps%22">Harmonic maps</searchLink><br /><searchLink fieldCode="DE" term="%22Tensor+fields%22">Tensor fields</searchLink><br /><searchLink fieldCode="DE" term="%22Artificial+neural+networks%22">Artificial neural networks</searchLink><br /><searchLink fieldCode="DE" term="%22Laplacian+matrices%22">Laplacian matrices</searchLink><br /><searchLink fieldCode="DE" term="%22Metric+spaces%22">Metric spaces</searchLink><br /><searchLink fieldCode="DE" term="%22Injective+functions%22">Injective functions</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We present a fast and robust method for computing an injective parameterization with low isometric distortion for disk‐like triangular meshes. Harmonic function‐based methods, with their rich mathematical foundation, are widely used. Harmonic maps are particularly valuable for ensuring injectivity under certain boundary conditions. In addition, they offer computational efficiency by forming a linear subspace [FW22]. However, this restricted subspace often leads to significant isometric distortion, especially for highly curved surfaces. Conversely, methods that operate in the full space of piecewise linear maps [SPSH*17] achieve lower isometric distortion, but at a higher computational cost. Aigerman et al. [AGK*22] pioneered a parameterization method that uses deep neural networks to predict the Jacobians of the map at mesh triangles, and integrates them into an explicit map by solving a Poisson equation. However, this approach often results in significant Poisson reconstruction errors due to the inability to ensure the integrability of the predicted neural Jacobian field, leading to unbounded distortion and lack of local injectivity. We propose a hybrid method that combines the speed and robustness of harmonic maps with the generality of deep neural networks to produce injective maps with low isometric distortion much faster than state‐of‐the‐art methods. The core concept is simple but powerful. Instead of learning Jacobian fields, we learn metric tensor fields over the input mesh, resulting in a cus‐tomized Laplacian matrix that defines a harmonic map in a modified metric [WGS23]. Our approach ensures injectivity, offers great computational efficiency, and produces significantly lower isometric distortion compared to straightforward harmonic maps. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Computer Graphics Forum is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1111/cgf.70061 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 14 StartPage: 1 Subjects: – SubjectFull: Harmonic maps Type: general – SubjectFull: Tensor fields Type: general – SubjectFull: Artificial neural networks Type: general – SubjectFull: Laplacian matrices Type: general – SubjectFull: Metric spaces Type: general – SubjectFull: Injective functions Type: general Titles: – TitleFull: Learning Metric Fields for Fast Low‐Distortion Mesh Parameterizations. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Fargion, G. – PersonEntity: Name: NameFull: Weber, O. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 05 Text: May2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 01677055 Numbering: – Type: volume Value: 44 – Type: issue Value: 2 Titles: – TitleFull: Computer Graphics Forum Type: main |
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