Dynamic off-the-grid untangling of curves by Riemannian metric
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| Format: | Preprint |
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2025
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| _version_ | 1866918091161600000 |
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| author | Laville, Bastien Bertrand, Théo |
| author_facet | Laville, Bastien Bertrand, Théo |
| contents | We propose an improved strategy for point sources tracking in a temporal stack through an off-the-grid fashion, inspired by the Benamou-Brenier regularisation in the literature. We define a lifting of the problem in the higher-dimensional space of the roto-translation group. This allows us to overcome the theoretical limitation of the off-the-grid method towards tangled point source trajectories, thus enabling the reconstruction and untangling even from the numerical standpoint. We define accordingly a new regularisation based on the relaxed Reeds-Shepp metric, an approximation of the sub-Riemannian Reeds-Shepp metric, further allowing control on the local curvature of the recovered trajectories. Then, we derive some properties of the discretisation and prove a $Γ$-convergence result, fostering interest for practical applications of polygonal, Bézier, and piecewise geodesic discretisation. We finally test our proposed method on a localisation problem example, and give a fair comparison with the state-of-the-art off-the-grid method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_10359 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamic off-the-grid untangling of curves by Riemannian metric Laville, Bastien Bertrand, Théo Differential Geometry Functional Analysis Optics 46E27, 49N45, 53C80, 53C21, 58E30, 34K29 We propose an improved strategy for point sources tracking in a temporal stack through an off-the-grid fashion, inspired by the Benamou-Brenier regularisation in the literature. We define a lifting of the problem in the higher-dimensional space of the roto-translation group. This allows us to overcome the theoretical limitation of the off-the-grid method towards tangled point source trajectories, thus enabling the reconstruction and untangling even from the numerical standpoint. We define accordingly a new regularisation based on the relaxed Reeds-Shepp metric, an approximation of the sub-Riemannian Reeds-Shepp metric, further allowing control on the local curvature of the recovered trajectories. Then, we derive some properties of the discretisation and prove a $Γ$-convergence result, fostering interest for practical applications of polygonal, Bézier, and piecewise geodesic discretisation. We finally test our proposed method on a localisation problem example, and give a fair comparison with the state-of-the-art off-the-grid method. |
| title | Dynamic off-the-grid untangling of curves by Riemannian metric |
| topic | Differential Geometry Functional Analysis Optics 46E27, 49N45, 53C80, 53C21, 58E30, 34K29 |
| url | https://arxiv.org/abs/2507.10359 |