Diffusion-Shock PDEs for Deep Learning on Position-Orientation Space

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.