# Learning to Discover Reflection Symmetry via Polar Matching Convolution

arXiv: https://arxiv.org/abs/2108.12952 · HTML: https://mlsanity.com/paper/2108.12952/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 42 references that are in the map.
- First posted on arXiv 30 Aug 2021.

## Builds on (most cited first, 12 of 12)

- [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)
- [DeepLab: Semantic Image Segmentation with Deep Convolutional Nets, Atrous Convolution, and Fully Connected CRFs](https://mlsanity.com/paper/1606.00915/) (2016)
- [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)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Beyond Planar Symmetry: Modeling human perception of reflection and rotation symmetries in the wild](https://mlsanity.com/paper/1704.03568/) (2017)
- [A convolutional approach to reflection symmetry](https://mlsanity.com/paper/1609.05257/) (2016)
- [Object Skeleton Extraction in Natural Images by Fusing Scale-associated Deep Side Outputs](https://mlsanity.com/paper/1603.09446/) (2016)
- [Multiple Reflection Symmetry Detection via Linear-Directional Kernel Density Estimation](https://mlsanity.com/paper/1704.06392/) (2017)

## Built on by (most cited first, 11 of 11)

- [Reflection and Rotation Symmetry Detection via Equivariant Learning](https://mlsanity.com/paper/2203.16787/) (2022)
- [Symmetry-Aware Transformer-based Mirror Detection](https://mlsanity.com/paper/2207.06332/) (2022)
- [NeRD++: Improved 3D-mirror symmetry learning from a single image](https://mlsanity.com/paper/2112.12579/) (2021)
- [Robust Extrinsic Symmetry Estimation in 3D Point Clouds](https://mlsanity.com/paper/2109.09927/) (2021)
- [SPA-VAE: Similar-Parts-Assignment for Unsupervised 3D Point Cloud Generation](https://mlsanity.com/paper/2203.07825/) (2022)
- [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)
- [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)
- [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 task of reflection symmetry detection remains challenging due to significant variations and ambiguities of symmetry patterns in the wild. Furthermore, since the local regions are required to match in reflection for detecting a symmetry pattern, it is hard for standard convolutional networks, which are not equivariant to rotation and reflection, to learn the task. To address the issue, we introduce a new convolutional technique, dubbed the polar matching convolution, which leverages a polar feature pooling, a self-similarity encoding, and a systematic kernel design for axes of different angles. The proposed high-dimensional kernel convolution network effectively learns to discover symmetry patterns from real-world images, overcoming the limitations of standard convolution. In addition, we present a new dataset and introduce a self-supervised learning strategy by augmenting the dataset with synthesizing images. Experiments demonstrate that our method outperforms state-of-the-art methods in terms of accuracy and robustness.