Diffusion-Shock Filtering on the Space of Positions and Orientations

Abstract

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.

Accepted in 10th International Conference on Scale Space and Variational Methods in Computer Vision

References in corpus (2)

• scikit-image: Image processing in Python
• Diffusion-Shock Inpainting