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We extend Regularised Diffusion-Shock (RDS) filtering from Euclidean space \(\mathbb {R}^2\) [26] 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 [9, 29], NLM [8], and BM3D [13] 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.

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Diffusion-Shock Filtering on the Space of Positions and Orientations

  • 会议论文

  • 发表日期：

    2025年

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  • Finn M. Sherry,
  • Kristina Schaefer,
  • Remco Duits

摘要

We extend Regularised Diffusion-Shock (RDS) filtering from Euclidean space \(\mathbb {R}^2\) [26] 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 [9, 29], NLM [8], and BM3D [13] 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.