Diffusion-Shock Filtering on the Space of Positions and Orientations

2025/02/24 by Sherry, Finn M., Schaefer, Kristina, Duits, Remco · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2502.17146

Abstract

We extend Regularised Diffusion-Shock (RDS) filtering from Euclidean space ℝ² to the space of positions and orientations \mathbbM₂ := ℝ² × S¹. This has numerous advantages, e.g. making it possible to enhance and inpaint crossing structures, since they become disentangled when lifted to \mathbbM₂. 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 \mathbbM₂ RDS inpainting is indeed able to restore crossing structures, unlike ℝ² RDS inpainting.

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