# Fourier Series Expansion Based Filter Parametrization for Equivariant Convolutions

arXiv: https://arxiv.org/abs/2107.14519 · HTML: https://mlsanity.com/paper/2107.14519/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 46 references that are in the map.
- First posted on arXiv 30 Jul 2021.

## Builds on (most cited first, 24 of 24)

- [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)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Enhanced Deep Residual Networks for Single Image Super-Resolution](https://mlsanity.com/paper/1707.02921/) (2017)
- [Image Super-Resolution Using Very Deep Residual Channel Attention Networks](https://mlsanity.com/paper/1807.02758/) (2018)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Residual Dense Network for Image Super-Resolution](https://mlsanity.com/paper/1802.08797/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Deep Learning for Image Super-resolution: A Survey](https://mlsanity.com/paper/1902.06068/) (2019)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Rotation equivariant vector field networks](https://mlsanity.com/paper/1612.09346/) (2016)
- [Polar Transformer Networks](https://mlsanity.com/paper/1709.01889/) (2017)
- [Oriented Response Networks](https://mlsanity.com/paper/1701.01833/) (2017)
- [TI-POOLING: transformation-invariant pooling for feature learning in Convolutional Neural Networks](https://mlsanity.com/paper/1604.06318/) (2016)
- [PDO-eConvs: Partial Differential Operator Based Equivariant Convolutions](https://mlsanity.com/paper/2007.10408/) (2020)
- [HexaConv](https://mlsanity.com/paper/1803.02108/) (2018)
- [PDO-eS2CNNs: Partial Differential Operator Based Equivariant Spherical CNNs](https://mlsanity.com/paper/2104.03584/) (2021)

## Built on by (most cited first, 26 of 26)

- [Improved Implicit Neural Representation with Fourier Reparameterized Training](https://mlsanity.com/paper/2401.07402/) (2024)
- [Rotation Equivariant Proximal Operator for Deep Unfolding Methods in Image Restoration](https://mlsanity.com/paper/2312.15701/) (2023)
- [Orientation-Shared Convolution Representation for CT Metal Artifact Learning](https://mlsanity.com/paper/2212.13166/) (2022)
- [MEPNet: A Model-Driven Equivariant Proximal Network for Joint Sparse-View Reconstruction and Metal Artifact Reduction in CT Images](https://mlsanity.com/paper/2306.14274/) (2023)
- [Rotation-Equivariant Self-Supervised Method in Image Denoising](https://mlsanity.com/paper/2505.19618/) (2025)
- [RSF-Conv: Rotation-and-Scale Equivariant Fourier Parameterized Convolution for Retinal Vessel Segmentation](https://mlsanity.com/paper/2309.15638/) (2023)
- [A Regularization-Guided Equivariant Approach for Image Restoration](https://mlsanity.com/paper/2505.19799/) (2025)
- [Rotation Equivariant Arbitrary-scale Image Super-Resolution](https://mlsanity.com/paper/2508.05160/) (2025)
- [SRE-CNN: A Spatiotemporal Rotation-Equivariant CNN for Cardiac Cine MR Imaging](https://mlsanity.com/paper/2409.08537/) (2024)
- [Vanilla Group Equivariant Vision Transformer: Simple and Effective](https://mlsanity.com/paper/2602.08047/) (2026)
- [Scale-Rotation-Equivariant Lie Group Convolution Neural Networks (Lie Group-CNNs)](https://mlsanity.com/paper/2306.06934/) (2023)
- [Hierarchical Invariance for Robust and Interpretable Vision Tasks at Larger Scales](https://mlsanity.com/paper/2402.15430/) (2024)
- [Rotation Equivariant Mamba for Vision Tasks](https://mlsanity.com/paper/2603.09138/) (2026)
- [Image-to-Image Translation Framework Embedded with Rotation Symmetry Priors](https://mlsanity.com/paper/2604.12805/) (2026)
- [TRG-Net: An Interpretable and Controllable Rain Generator](https://mlsanity.com/paper/2403.09993/) (2024)
- [Feature Alignment with Equivariant Convolutions for Burst Image Super-Resolution](https://mlsanity.com/paper/2503.08300/) (2025)
- [Continuous Filtered Backprojection by Learnable Interpolation Network](https://mlsanity.com/paper/2505.01768/) (2025)
- [DUN-SRE: Deep Unrolling Network with Spatiotemporal Rotation Equivariance for Dynamic MRI Reconstruction](https://mlsanity.com/paper/2506.10309/) (2025)
- [Progressive Alignment Degradation Learning for Pansharpening](https://mlsanity.com/paper/2506.20179/) (2025)
- [Hardware-Friendly Input Expansion for Accelerating Function Approximation](https://mlsanity.com/paper/2602.17952/) (2026)
- [Neural Approximation and Its Applications](https://mlsanity.com/paper/2603.13311/) (2026)
- [ReDiffuse: Rotation Equivariant Diffusion Model for Multi-focus Image Fusion](https://mlsanity.com/paper/2603.21129/) (2026)
- [Aligning Network Equivariance with Data Symmetry: A Theoretical Framework and Adaptive Approach for Image Restoration](https://mlsanity.com/paper/2605.13744/) (2026)
- [Flash EQ-Linear: Accelerating Equivariant Linear Layers via Group-wise Discrete Fourier Transform](https://mlsanity.com/paper/2607.21271/) (2026)
- [Embedding Rotation Invariance for Provable Multi-Oriented Scene Text Recognition](https://mlsanity.com/paper/2608.10684/) (2026)
- [RoES: Rotational Equivariant Selective-frequency Fusion for Multimodal Images](https://mlsanity.com/paper/2609.12497/) (2026)

## Abstract (arXiv)

It has been shown that equivariant convolution is very helpful for many types of computer vision tasks. Recently, the 2D filter parametrization technique plays an important role when designing equivariant convolutions. However, the current filter parametrization method still has its evident drawbacks, where the most critical one lies in the accuracy problem of filter representation. Against this issue, in this paper we modify the classical Fourier series expansion for 2D filters, and propose a new set of atomic basis functions for filter parametrization. The proposed filter parametrization method not only finely represents 2D filters with zero error when the filter is not rotated, but also substantially alleviates the fence-effect-caused quality degradation when the filter is rotated. Accordingly, we construct a new equivariant convolution method based on the proposed filter parametrization method, named F-Conv. We prove that the equivariance of the proposed F-Conv is exact in the continuous domain, which becomes approximate only after discretization. Extensive experiments show the superiority of the proposed method. Particularly, we adopt rotation equivariant convolution methods to image super-resolution task, and F-Conv evidently outperforms previous filter parametrization based method in this task, reflecting its intrinsic capability of faithfully preserving rotation symmetries in local image features.