Papers › Diffusion-Shock Filtering on the Space of Positions and Orientations

Diffusion-Shock Filtering on the Space of Positions and Orientations

24 Feb 2025arXiv:2502.17146links table onlyarchive 2025-07-28

Finn M. Sherry, Kristina Schaefer, Remco Duits

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We extend Regularised Diffusion-Shock (RDS) filtering from Euclidean space ℝ² to the space of positions and orientations 𝕄₂ := ℝ² ×S¹. This has numerous advantages, e.g. making it possible to enhance and inpaint crossing structures, since they become disentangled when lifted to 𝕄₂. 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 𝕄₂ RDS inpainting is indeed able to restore crossing structures, unlike ℝ² RDS inpainting.

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