# SO(2) and O(2) Equivariance in Image Recognition with Bessel-Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2304.09214 · HTML: https://mlsanity.com/paper/2304.09214/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 32 references that are in the map.
- First posted on arXiv 18 Apr 2023.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Deformable Convolutional Networks](https://mlsanity.com/paper/1703.06211/) (2017)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Rotation-invariant convolutional neural networks for galaxy morphology prediction](https://mlsanity.com/paper/1503.07077/) (2015)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Scale-Invariant Convolutional Neural Networks](https://mlsanity.com/paper/1411.6369/) (2014)
- [Dense Steerable Filter CNNs for Exploiting Rotational Symmetry in Histology Images](https://mlsanity.com/paper/2004.03037/) (2020)
- [Learning rotation invariant convolutional filters for texture classification](https://mlsanity.com/paper/1604.06720/) (2016)
- [Scale Steerable Filters for Locally Scale-Invariant Convolutional Neural Networks](https://mlsanity.com/paper/1906.03861/) (2019)
- [Multi-Instance Multi-Scale CNN for Medical Image Classification](https://mlsanity.com/paper/1907.02413/) (2019)
- [Attentive Normalization](https://mlsanity.com/paper/1908.01259/) (2019)
- [A Novel CNN-based Method for Accurate Ship Detection in HR Optical Remote Sensing Images via Rotated Bounding Box](https://mlsanity.com/paper/2004.07124/) (2020)
- [Deformation Robust Roto-Scale-Translation Equivariant CNNs](https://mlsanity.com/paper/2111.10978/) (2021)
- [Exploiting SIFT Descriptor for Rotation Invariant Convolutional Neural Network](https://mlsanity.com/paper/1904.00197/) (2019)

## Built on by (most cited first, 1 of 1)

- [On the effectiveness of Rotation-Equivariance in U-Net: A Benchmark for Image Segmentation](https://mlsanity.com/paper/2412.09182/) (2024)

## Abstract (arXiv)

For many years, it has been shown how much exploiting equivariances can be beneficial when solving image analysis tasks. For example, the superiority of convolutional neural networks (CNNs) compared to dense networks mainly comes from an elegant exploitation of the translation equivariance. Patterns can appear at arbitrary positions and convolutions take this into account to achieve translation invariant operations through weight sharing. Nevertheless, images often involve other symmetries that can also be exploited. It is the case of rotations and reflections that have drawn particular attention and led to the development of multiple equivariant CNN architectures. Among all these methods, Bessel-convolutional neural networks (B-CNNs) exploit a particular decomposition based on Bessel functions to modify the key operation between images and filters and make it by design equivariant to all the continuous set of planar rotations. In this work, the mathematical developments of B-CNNs are presented along with several improvements, including the incorporation of reflection and multi-scale equivariances. Extensive study is carried out to assess the performances of B-CNNs compared to other methods. Finally, we emphasize the theoretical advantages of B-CNNs by giving more insights and in-depth mathematical details.