# Axis-level Symmetry Detection with Group-Equivariant Representation

arXiv: https://arxiv.org/abs/2508.10740 · HTML: https://mlsanity.com/paper/2508.10740/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 62 references that are in the map.
- First posted on arXiv 14 Aug 2025.

## Builds on (most cited first, 30 of 30)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Faster R-CNN: Towards Real-Time Object Detection with Region Proposal Networks](https://mlsanity.com/paper/1506.01497/) (2015)
- [SSD: Single Shot MultiBox Detector](https://mlsanity.com/paper/1512.02325/) (2015)
- [Deformable Convolutional Networks](https://mlsanity.com/paper/1703.06211/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [ReDet: A Rotation-equivariant Detector for Aerial Object Detection](https://mlsanity.com/paper/2103.07733/) (2021)
- [Rotation equivariant vector field networks](https://mlsanity.com/paper/1612.09346/) (2016)
- [Rotation-invariant convolutional neural networks for galaxy morphology prediction](https://mlsanity.com/paper/1503.07077/) (2015)
- [End-to-End Wireframe Parsing](https://mlsanity.com/paper/1905.03246/) (2019)
- [Learning to Parse Wireframes in Images of Man-Made Environments](https://mlsanity.com/paper/2007.07527/) (2020)
- [Holistically-Attracted Wireframe Parsing](https://mlsanity.com/paper/2003.01663/) (2020)
- [Equivariant plug-and-play image reconstruction](https://mlsanity.com/paper/2312.01831/) (2023)
- [Steerers: A framework for rotation equivariant keypoint descriptors](https://mlsanity.com/paper/2312.02152/) (2023)
- [ELSD: Efficient Line Segment Detector and Descriptor](https://mlsanity.com/paper/2104.14205/) (2021)
- [Beyond Planar Symmetry: Modeling human perception of reflection and rotation symmetries in the wild](https://mlsanity.com/paper/1704.03568/) (2017)
- [Intertwiners between Induced Representations (with Applications to the Theory of Equivariant Neural Networks)](https://mlsanity.com/paper/1803.10743/) (2018)
- [Robust Symmetry Detection via Riemannian Langevin Dynamics](https://mlsanity.com/paper/2410.02786/) (2024)
- [Reflection and Rotation Symmetry Detection via Equivariant Learning](https://mlsanity.com/paper/2203.16787/) (2022)
- [Learning to Discover Reflection Symmetry via Polar Matching Convolution](https://mlsanity.com/paper/2108.12952/) (2021)
- [Symmetry Strikes Back: From Single-Image Symmetry Detection to 3D Generation](https://mlsanity.com/paper/2411.17763/) (2024)
- [SE(3) Equivariant Ray Embeddings for Implicit Multi-View Depth Estimation](https://mlsanity.com/paper/2411.07326/) (2024)
- [3D Equivariant Pose Regression via Direct Wigner-D Harmonics Prediction](https://mlsanity.com/paper/2411.00543/) (2024)
- [Leveraging 3D Geometric Priors in 2D Rotation Symmetry Detection](https://mlsanity.com/paper/2503.20235/) (2025)

## Built on by (most cited first, 2 of 2)

- [Quantum-Inspired Harmonic Decision Models: A Computational Framework for Music Generation](https://mlsanity.com/paper/2607.05007/) (2026)
- [Classical Versus Deep Mirror-Symmetry Scoring: A Benchmark of Thirteen Methods](https://mlsanity.com/paper/2607.08379/) (2026)

## Abstract (arXiv)

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.