Axis-level Symmetry Detection with Group-Equivariant Representation

TL;DR

This work tackles axis-level symmetry detection in complex scenes by modeling reflection as lines and rotation as points within a dihedral group $ ext{D}_N$-equivariant representation. It introduces a dual-branch network with orientational anchors and dedicated reflectional/rotational matching to predict precise axis parameters, achieving robustness to rotations and reflections. Empirical results on the DENDI and related datasets show state-of-the-art performance for both reflection and rotation symmetry detection, with ablations validating the effectiveness of orientational anchors and group-aware matching. The approach provides a principled framework for symmetry understanding with practical benefits for scene analysis and robotics, and it points to future extensions to continuous groups, 3D spaces, and viewpoint variations.

Abstract

Symmetry is a fundamental concept that has been extensively studied, yet detecting it in complex scenes remains a significant challenge in computer vision. Recent heatmap-based approaches can localize potential regions of symmetry axes but often lack precision in identifying individual axes. In this work, we propose a novel framework for axis-level detection of the two most common symmetry types-reflection and rotation-by representing them as explicit geometric primitives, i.e. lines and points. Our method employs a dual-branch architecture that is equivariant to the dihedral group, with each branch specialized to exploit the structure of dihedral group-equivariant features for its respective symmetry type. For reflection symmetry, we introduce orientational anchors, aligned with group components, to enable orientation-specific detection, and a reflectional matching that measures similarity between patterns and their mirrored counterparts across candidate axes. For rotational symmetry, we propose a rotational matching that compares patterns at fixed angular intervals to identify rotational centers. Extensive experiments demonstrate that our method achieves state-of-the-art performance, outperforming existing approaches.

Figures (7)

• Figure 1: Comparison of the heatmap-based method seo2022equisym and our axis-level approach on rotated inputs. The red triangle indicates the rotated orientation of input image. Our axis-level symmetry detection method captures reflection (green lines) and rotation (red points) axes as precise geometric entities and demonstrates superior robustness to rotation compared to the heatmap-based method.
• Figure 2: Overall architecture of our proposed instance-level symmetry detection network. Given an input image, a $\mathrm{D}_N$-equivariant backbone extracts features $\mathbf{F} \in \mathbb{R}^{H \times W \times \mathcal{C} |\mathrm{D}_N|}$. The reflection branch (top) employs equivariant reflectional matching and orientational anchor expansion to predict reflection axes as parameterized line segments $(\alpha, x, y, p, \rho, \theta)$. The rotation branch (bottom) applies invariant rotational matching to detect rotation axes and classify their fold classes parameterized as $(x,y,p_s)$.
• Figure 3: Illustration of our orientational anchor expansion on ${\mathrm{\mathbf D}_8}$ group. The $\mathrm{D}_8$-equivariant features $\mathbf{Y}_\kappa$ undergo transformation $\mathcal{N}_\kappa$ and aggregation $\circledast_\kappa$, creating $\mathrm{C}_8$-equivariant features $\mathbf{O}_\kappa$. These are combined across opposite orientations to handle the $\theta$ and $\theta + \pi$ equivalence, allowing each orientation channel to specialize in specific angular ranges and improve detection of axes with overlapping midpoints. Each arrow represents a feature map.
• Figure 4: Illustration of our equivariant reflectional matching (left) and invariant rotational matching (right) modules. The reflectional matching computes similarity scores between rotated features and their reflections across all $|\mathrm{C}_N|$ rotation angles, preserving dihedral group equivariance with rotation invariance. The rotational matching computes similarities between feature pairs with different rotation angle interval, yielding rotation-invariant features for detecting $n$-fold rotation symmetry centers. Both modules incorporate spatial neighborhoods $\mathcal{Q}_k$ for robust detection across multiple scales.
• Figure 5: Qualitative comparison of symmetry detection methods. Our instance-wise approach produces clearer, more precise symmetry instances compared to heatmap-based methods seoshim2021pmcnetseo2022equisym, especially for smaller objects and complex scenes. Green lines in ground truth and our results represent reflection axes, while red points represent rotation axes.