Diffusion-Shock PDEs for Deep Learning on Position-Orientation Space
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912974045708288 |
|---|---|
| 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 $\mathbb{R}_2$ [1] to position-orientation space $\mathbb{M}_2 \cong \mathbb{R}^2 \times S^1$. 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. Furthermore, we see that $\mathbb{M}_2$ RDS inpainting is indeed able to restore crossing structures, unlike $\mathbb{R}^2$ RDS inpainting. In addition to the contributions of our SSVM submission "Diffusion-Shock Filtering on the Space of Positions and Orientations", in this extended work we provide new theorical results and automate RDS filtering by integrating it into a geometric deep learning framework. Regarding our theoretical contributions, we prove that our generalised diffusions are still well-posed, smoothing, and analytic. We developed an RDS filtering PDE layer for the PDE-CNN and PDE-G-CNN deep learning frameworks, using a novel gating mechanism. We show that these new RDS PDE layers can be beneficial in various impainting and denoising tasks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06405 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diffusion-Shock PDEs for Deep Learning on Position-Orientation Space Sherry, Finn M. Schaefer, Kristina Duits, Remco Differential Geometry We extend Regularised Diffusion-Shock (RDS) filtering from Euclidean space $\mathbb{R}_2$ [1] to position-orientation space $\mathbb{M}_2 \cong \mathbb{R}^2 \times S^1$. 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. Furthermore, we see that $\mathbb{M}_2$ RDS inpainting is indeed able to restore crossing structures, unlike $\mathbb{R}^2$ RDS inpainting. In addition to the contributions of our SSVM submission "Diffusion-Shock Filtering on the Space of Positions and Orientations", in this extended work we provide new theorical results and automate RDS filtering by integrating it into a geometric deep learning framework. Regarding our theoretical contributions, we prove that our generalised diffusions are still well-posed, smoothing, and analytic. We developed an RDS filtering PDE layer for the PDE-CNN and PDE-G-CNN deep learning frameworks, using a novel gating mechanism. We show that these new RDS PDE layers can be beneficial in various impainting and denoising tasks. |
| title | Diffusion-Shock PDEs for Deep Learning on Position-Orientation Space |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2509.06405 |