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Diffusion-Shock Filtering on the Space of Positions and Orientations

Finn M. Sherry, Kristina Schaefer, Remco Duits

TL;DR

This work extends Regularised Diffusion-Shock Filtering (RDS) from the Euclidean plane to the space of positions and orientations $\mathbb{M}_2 = \mathbb{R}^2 \times S^1$, enabling crossing-preserving denoising and inpainting via an orientation-score lifting with cake wavelets. It introduces two PDE-based variants on $\mathbb{M}_2$—left-invariant RDS and gauge-frame RDS—and analyzes the associated generalized Laplacians: in the left-invariant setting the diffusion is governed by the Lie-Cartan/Laplace-Beltrami operators, while the gauge-frame case yields a data-driven Lie-Cartan Laplacian. Theoretical results show equivalence of the Lie-Cartan and LB operators in the unimodular setting but reveal a discrepancy in the gauge-frame construction, and experiments demonstrate superior PSNR performance and crossing restoration on $\mathbb{M}_2$ compared to TR-TV, BM3D, and NLM. The work provides publicly available code and paves the way for integrating crossing-aware diffusion-shock processing with geometric PDEs and PDE-based deep learning for multi-orientation image analysis.

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.

Diffusion-Shock Filtering on the Space of Positions and Orientations

TL;DR

This work extends Regularised Diffusion-Shock Filtering (RDS) from the Euclidean plane to the space of positions and orientations , enabling crossing-preserving denoising and inpainting via an orientation-score lifting with cake wavelets. It introduces two PDE-based variants on —left-invariant RDS and gauge-frame RDS—and analyzes the associated generalized Laplacians: in the left-invariant setting the diffusion is governed by the Lie-Cartan/Laplace-Beltrami operators, while the gauge-frame case yields a data-driven Lie-Cartan Laplacian. Theoretical results show equivalence of the Lie-Cartan and LB operators in the unimodular setting but reveal a discrepancy in the gauge-frame construction, and experiments demonstrate superior PSNR performance and crossing restoration on compared to TR-TV, BM3D, and NLM. The work provides publicly available code and paves the way for integrating crossing-aware diffusion-shock processing with geometric PDEs and PDE-based deep learning for multi-orientation image analysis.

Abstract

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

Paper Structure

This paper contains 7 sections, 1 theorem, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our Contributions.
  3. Regularised Diffusion-Shock Filtering on Euclidean Space
  4. Space of Positions & Orientations
  5. Regularised Diffusion-Shock Filtering on Space of Positions and Orientations
  6. Gauge Frames.
  7. Generalised Laplacians

Key Result

theorem thmcountertheorem

Let $G$ be a connected Lie group, let $\mathcal{G}$ be a left-invariant metric tensor field thereon, and let $\nu \in \mathbb{R}$. With respect to a left-invariant frame $\{\mathcal{A}_i\}_i$, the Lie-Cartan Laplacian is given by while the Laplace-Beltrami operator is given by with $c_{ij}^k$ the structure constants defined by $c_{ij}^k \mathcal{A}_k = [\mathcal{A}_i, \mathcal{A}_j]$. The differ

Figures (2)

  • Figure 1: (a) Cake wavelet for orientation $\theta = \frac{\pi}{8}$. (b) Lifting disentangles crossing and overlapping structures.
  • Figure 2: Comparison of gauge frame and standard left-invariant frame. From the top view (a) we see that $\mathcal{A}_1^U$ has been rotated towards $\mathcal{A}_2$ to compensate for the deviation from horizontality $d_H$. From the side view (b) we see that $\mathcal{A}_1^U$ has been rotated towards $\mathcal{A}_3$; the rotation angle is related to the curvature $\kappa$.

Theorems & Definitions (7)

  • definition thmcounterdefinition: Space of positions and orientations
  • definition thmcounterdefinition: Left-Invariant Frame
  • definition thmcounterdefinition: Orientation Score
  • definition thmcounterdefinition: 1st Gauge Vector
  • definition thmcounterdefinition: Generalised Laplacian
  • definition thmcounterdefinition: Lie-Cartan Connection
  • theorem thmcountertheorem: Lie-Cartan Laplacians