Axis-level Symmetry Detection with Group-Equivariant Representation

2025-08-14

AI Summary

This paper addresses the precise localization of reflection and rotational symmetry axes in complex scenes. Methodologically, it proposes an equivariant dual-branch detection framework tailored for dihedral groups (Dₙ): (1) symmetry axes are explicitly modeled as geometric primitives; (2) a dual-branch architecture separately handles reflection and rotational symmetries, incorporating directional anchor alignment, reflection matching, and angular-interval–based rotational matching; (3) joint optimization integrates group-equivariant representation learning, heatmap-based localization, and geometric priors. Evaluated on multiple benchmarks, the method achieves state-of-the-art performance, significantly improving both axis localization accuracy and cross-scene robustness. It is the first approach to enable unified, interpretable, and high-precision detection of axis-level symmetries.

Technology Category

• Computer Vision: Visual Reasoning & Symbolic Representations
• Intelligent Robots: Multimodal Perception & Sensor Fusion
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Security and Privacy: Data transparency and provenance

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.

Problem

• Detect reflection and rotation symmetry axes precisely
• Overcome limitations of heatmap-based symmetry detection methods
• Improve accuracy in identifying individual symmetry axes

Innovation

• Dual-branch dihedral group-equivariant architecture
• Orientational anchors for reflection symmetry
• Rotational matching for symmetry center detection