Finsler-Laplace-Beltrami Operators with Application to Shape Analysis
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909161953951744 |
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| author | Weber, Simon Dagès, Thomas Gao, Maolin Cremers, Daniel |
| author_facet | Weber, Simon Dagès, Thomas Gao, Maolin Cremers, Daniel |
| contents | The Laplace-Beltrami operator (LBO) emerges from studying manifolds equipped with a Riemannian metric. It is often called the Swiss army knife of geometry processing as it allows to capture intrinsic shape information and gives rise to heat diffusion, geodesic distances, and a multitude of shape descriptors. It also plays a central role in geometric deep learning. In this work, we explore Finsler manifolds as a generalization of Riemannian manifolds. We revisit the Finsler heat equation and derive a Finsler heat kernel and a Finsler-Laplace-Beltrami Operator (FLBO): a novel theoretically justified anisotropic Laplace-Beltrami operator (ALBO). In experimental evaluations we demonstrate that the proposed FLBO is a valuable alternative to the traditional Riemannian-based LBO and ALBOs for spatial filtering and shape correspondence estimation. We hope that the proposed Finsler heat kernel and the FLBO will inspire further exploration of Finsler geometry in the computer vision community. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_03999 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finsler-Laplace-Beltrami Operators with Application to Shape Analysis Weber, Simon Dagès, Thomas Gao, Maolin Cremers, Daniel Computer Vision and Pattern Recognition The Laplace-Beltrami operator (LBO) emerges from studying manifolds equipped with a Riemannian metric. It is often called the Swiss army knife of geometry processing as it allows to capture intrinsic shape information and gives rise to heat diffusion, geodesic distances, and a multitude of shape descriptors. It also plays a central role in geometric deep learning. In this work, we explore Finsler manifolds as a generalization of Riemannian manifolds. We revisit the Finsler heat equation and derive a Finsler heat kernel and a Finsler-Laplace-Beltrami Operator (FLBO): a novel theoretically justified anisotropic Laplace-Beltrami operator (ALBO). In experimental evaluations we demonstrate that the proposed FLBO is a valuable alternative to the traditional Riemannian-based LBO and ALBOs for spatial filtering and shape correspondence estimation. We hope that the proposed Finsler heat kernel and the FLBO will inspire further exploration of Finsler geometry in the computer vision community. |
| title | Finsler-Laplace-Beltrami Operators with Application to Shape Analysis |
| topic | Computer Vision and Pattern Recognition |
| url | https://arxiv.org/abs/2404.03999 |