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

作者：Finn M. Sherry, Kristina Schaefer, R. Duits · 发表于：Journal of Mathematical Imaging and Vision · 年份：2025 · DOI：10.1007/s10851-026-01291-z · 研究领域：Computer Science、Mathematics、Medicine

We extend regularised diffusion–shock (RDS) filtering from Euclidean space R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document} (Schaefer and Weickert in J Math Imaging Vis 66:447–463, 2024. https://doi.org/10.1007/s10851-024-01175-0) to position–orientation space M2≅R2×S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {M}_2\cong \mathbb {R}^2\times S^1$$\end{document}. This has numerous advantages, e.g. making it possible to enhance and inpaint crossing structures, since they become disentangled when lifted to M2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {M}_2$$\end{document}. 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...