Diffusion-Shock Filtering on the Space of Positions and Orientations
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909620661911552 |
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| author | Sherry, Finn M. Schaefer, Kristina Duits, Remco |
| author_facet | Sherry, Finn M. Schaefer, Kristina Duits, Remco |
| contents | We extend Regularised Diffusion-Shock (RDS) filtering from Euclidean space $\mathbb{R}^2$ to the space of positions and orientations $\mathbb{M}_2 := \mathbb{R}^2 \times S^1$. This has numerous advantages, e.g. making it possible to enhance and inpaint crossing structures, since they become disentangled when lifted to $\mathbb{M}_2$. We create a version of the algorithm using gauge frames to mitigate issues caused by lifting to a finite number of orientations. This leads us to study generalisations of diffusion, since the gauge frame diffusion is not generated by the Laplace-Beltrami operator. RDS filtering compares favourably to existing techniques such as Total Roto-Translational Variation (TR-TV) flow, NLM, and BM3D when denoising images with crossing structures, particularly if they are segmented. Additionally, we see that $\mathbb{M}_2$ RDS inpainting is indeed able to restore crossing structures, unlike $\mathbb{R}^2$ RDS inpainting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17146 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diffusion-Shock Filtering on the Space of Positions and Orientations Sherry, Finn M. Schaefer, Kristina Duits, Remco Differential Geometry We extend Regularised Diffusion-Shock (RDS) filtering from Euclidean space $\mathbb{R}^2$ to the space of positions and orientations $\mathbb{M}_2 := \mathbb{R}^2 \times S^1$. This has numerous advantages, e.g. making it possible to enhance and inpaint crossing structures, since they become disentangled when lifted to $\mathbb{M}_2$. We create a version of the algorithm using gauge frames to mitigate issues caused by lifting to a finite number of orientations. This leads us to study generalisations of diffusion, since the gauge frame diffusion is not generated by the Laplace-Beltrami operator. RDS filtering compares favourably to existing techniques such as Total Roto-Translational Variation (TR-TV) flow, NLM, and BM3D when denoising images with crossing structures, particularly if they are segmented. Additionally, we see that $\mathbb{M}_2$ RDS inpainting is indeed able to restore crossing structures, unlike $\mathbb{R}^2$ RDS inpainting. |
| title | Diffusion-Shock Filtering on the Space of Positions and Orientations |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2502.17146 |