Robust Symmetry Detection via Riemannian Langevin Dynamics

TL;DR

The paper targets robust detection of reflective symmetries in noisy 2D and 3D shapes by marrying classical transformation-space clustering with score-based diffusion ideas. It reframes symmetry detection as mode-seeking on a bounded transformation space using Langevin dynamics on a Riemannian manifold, with a geodesic distance guiding updates. The approach yields improved robustness to noise, detects both partial and global symmetries, and supports downstream tasks such as shape fixing and compression; it also demonstrates extensions to other symmetry types. Overall, the method provides a training-free, geometry-grounded framework that leverages modern diffusion concepts to enhance symmetry detection in practical pipelines.

Abstract

Symmetries are ubiquitous across all kinds of objects, whether in nature or in man-made creations. While these symmetries may seem intuitive to the human eye, detecting them with a machine is nontrivial due to the vast search space. Classical geometry-based methods work by aggregating "votes" for each symmetry but struggle with noise. In contrast, learning-based methods may be more robust to noise, but often overlook partial symmetries due to the scarcity of annotated data. In this work, we address this challenge by proposing a novel symmetry detection method that marries classical symmetry detection techniques with recent advances in generative modeling. Specifically, we apply Langevin dynamics to a redefined symmetry space to enhance robustness against noise. We provide empirical results on a variety of shapes that suggest our method is not only robust to noise, but can also identify both partial and global symmetries. Moreover, we demonstrate the utility of our detected symmetries in various downstream tasks, such as compression and symmetrization of noisy shapes.

Figures (13)

• Figure 1: Reflection across a hyperplane
• Figure 2: Transformation space construction
• Figure 3: Geodesic distance to the black point in transformation space. Blue indicates proximity, and red indicates larger distance. Shortest distance can be computed traveling through the origin as well.
• Figure 4: Stepping in the transformation space. We move both $T$ and $-T$, and only send the point to the valid destination.
• Figure 5: Visualizing Langevin dynamics trajectory. For each converged mode, we show the full trajectory (top row), as well as the corresponding symmetry in the shape space (bottom row). Note that due to our Langevin construction, we directly jump across the invalid region as opposed to stepping within it. When converged modes are near the boundary of the invalid region ($l \approx k$), we see oscillations through the origin as the two modes are equivalent ($H(0, \mathbf{n}) = H(0, -\mathbf{n})$).