# Reflection and Rotation Symmetry Detection via Equivariant Learning

arXiv: https://arxiv.org/abs/2203.16787 · HTML: https://mlsanity.com/paper/2203.16787/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 48 references that are in the map.
- First posted on arXiv 31 Mar 2022.

## Builds on (most cited first, 21 of 21)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Rethinking Atrous Convolution for Semantic Image Segmentation](https://mlsanity.com/paper/1706.05587/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Focal Loss for Dense Object Detection](https://mlsanity.com/paper/1708.02002/) (2017)
- [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)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [Rotation equivariant vector field networks](https://mlsanity.com/paper/1612.09346/) (2016)
- [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)
- [Rotation Equivariant Siamese Networks for Tracking](https://mlsanity.com/paper/2012.13078/) (2020)
- [Learning to Discover Reflection Symmetry via Polar Matching Convolution](https://mlsanity.com/paper/2108.12952/) (2021)
- [A convolutional approach to reflection symmetry](https://mlsanity.com/paper/1609.05257/) (2016)

## Built on by (most cited first, 13 of 13)

- [H2RBox-v2: Incorporating Symmetry for Boosting Horizontal Box Supervised Oriented Object Detection](https://mlsanity.com/paper/2304.04403/) (2023)
- [E2PN: Efficient SE(3)-Equivariant Point Network](https://mlsanity.com/paper/2206.05398/) (2022)
- [Canonical Fields: Self-Supervised Learning of Pose-Canonicalized Neural Fields](https://mlsanity.com/paper/2212.02493/) (2022)
- [SYM3D: Learning Symmetric Triplanes for Better 3D-Awareness of GANs](https://mlsanity.com/paper/2406.06432/) (2024)
- [Leveraging 3D Geometric Priors in 2D Rotation Symmetry Detection](https://mlsanity.com/paper/2503.20235/) (2025)
- [Axis-level Symmetry Detection with Group-Equivariant Representation](https://mlsanity.com/paper/2508.10740/) (2025)
- [CLIPSym: Delving into Symmetry Detection with CLIP](https://mlsanity.com/paper/2508.14197/) (2025)
- [Improving VisNet for Object Recognition](https://mlsanity.com/paper/2511.08897/) (2025)
- [Quantum-Inspired Harmonic Decision Models: A Computational Framework for Music Generation](https://mlsanity.com/paper/2607.05007/) (2026)
- [Symmetric Perception and Ordinal Regression for Detecting Scoliosis Natural Image](https://mlsanity.com/paper/2411.15799/) (2024)
- [Classical Versus Deep Mirror-Symmetry Scoring: A Benchmark of Thirteen Methods](https://mlsanity.com/paper/2607.08379/) (2026)
- [Water Reflection Detection Using Symmetric Attention](https://mlsanity.com/paper/2607.10749/) (2026)
- [ARD-REFSM: Enhancing Reflection Symmetry Detection with Asymmetric Denoising and Rotation Equivariance](https://mlsanity.com/paper/2607.27927/) (2026)

## Abstract (arXiv)

The inherent challenge of detecting symmetries stems from arbitrary orientations of symmetry patterns; a reflection symmetry mirrors itself against an axis with a specific orientation while a rotation symmetry matches its rotated copy with a specific orientation. Discovering such symmetry patterns from an image thus benefits from an equivariant feature representation, which varies consistently with reflection and rotation of the image. In this work, we introduce a group-equivariant convolutional network for symmetry detection, dubbed EquiSym, which leverages equivariant feature maps with respect to a dihedral group of reflection and rotation. The proposed network is built end-to-end with dihedrally-equivariant layers and trained to output a spatial map for reflection axes or rotation centers. We also present a new dataset, DENse and DIverse symmetry (DENDI), which mitigates limitations of existing benchmarks for reflection and rotation symmetry detection. Experiments show that our method achieves the state of the arts in symmetry detection on LDRS and DENDI datasets.