# RIC-CNN: Rotation-Invariant Coordinate Convolutional Neural Network

arXiv: https://arxiv.org/abs/2211.11812 · HTML: https://mlsanity.com/paper/2211.11812/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 56 references that are in the map.
- First posted on arXiv 21 Nov 2022.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [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)
- [Object Detection in 20 Years: A Survey](https://mlsanity.com/paper/1905.05055/) (2019)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [AID: A Benchmark Dataset for Performance Evaluation of Aerial Scene Classification](https://mlsanity.com/paper/1608.05167/) (2016)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [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)
- [Exploiting Cyclic Symmetry in Convolutional Neural Networks](https://mlsanity.com/paper/1602.02660/) (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)
- [Warped Convolutions: Efficient Invariance to Spatial Transformations](https://mlsanity.com/paper/1609.04382/) (2016)
- [Deep Rotation Equivariant Network](https://mlsanity.com/paper/1705.08623/) (2017)
- [Convolution with even-sized kernels and symmetric padding](https://mlsanity.com/paper/1903.08385/) (2019)
- [Understanding when spatial transformer networks do not support invariance, and what to do about it](https://mlsanity.com/paper/2004.11678/) (2020)

## Built on by (most cited first, 21 of 21)

- [SymmPI: Predictive Inference for Data with Group Symmetries](https://mlsanity.com/paper/2312.16160/) (2023)
- [Using and Abusing Equivariance](https://mlsanity.com/paper/2308.11316/) (2023)
- [3D Point Cloud Network Pruning: When Some Weights Do not Matter](https://mlsanity.com/paper/2408.14601/) (2024)
- [SRE-Conv: Symmetric Rotation Equivariant Convolution for Biomedical Image Classification](https://mlsanity.com/paper/2501.09753/) (2025)
- [Producing Plankton Classifiers that are Robust to Dataset Shift](https://mlsanity.com/paper/2401.14256/) (2024)
- [Achieving Rotation-Invariant Convolution via Non-Learnable Orientation Alignment Operators](https://mlsanity.com/paper/2404.11309/) (2024)
- [PreCM: The Padding-based Rotation Equivariant Convolution Mode for Semantic Segmentation](https://mlsanity.com/paper/2411.01624/) (2024)
- [GMR-Conv: An Efficient Rotation and Reflection Equivariant Convolution Kernel Using Gaussian Mixture Rings](https://mlsanity.com/paper/2504.02819/) (2025)
- [Normalize Filters! Classical Wisdom for Deep Vision](https://mlsanity.com/paper/2506.04401/) (2025)
- [Sorted Convolutional Network for Achieving Continuous Rotational Invariance](https://mlsanity.com/paper/2305.14462/) (2023)
- [Efficient 3D affinely equivariant CNNs with adaptive fusion of augmented spherical Fourier-Bessel bases](https://mlsanity.com/paper/2402.16825/) (2024)
- [MARs: Multi-view Attention Regularizations for Patch-based Feature Recognition of Space Terrain](https://mlsanity.com/paper/2410.05182/) (2024)
- [Rethinking Rotation-Invariant Recognition of Fine-grained Shapes from the Perspective of Contour Points](https://mlsanity.com/paper/2503.10992/) (2025)
- [Equivariant Imaging Biomarkers for Robust Unsupervised Segmentation of Histopathology](https://mlsanity.com/paper/2505.05689/) (2025)
- [Accelerated Rotation-Invariant Convolution for UAV Image Segmentation](https://mlsanity.com/paper/2512.08888/) (2025)
- [Equi-ViT: Rotational Equivariant Vision Transformer for Robust Histopathology Analysis](https://mlsanity.com/paper/2601.09130/) (2026)
- [Computing a Characteristic Orientation for Rotation-Independent Image Analysis](https://mlsanity.com/paper/2602.20930/) (2026)
- [Ctrl-A: Control-Driven Online Data Augmentation](https://mlsanity.com/paper/2603.21819/) (2026)
- [SPECSIA: Stylization Dataset for Novel-View Enhancement in Drawing-based 3D Animation](https://mlsanity.com/paper/2607.00525/) (2026)
- [4DMulti: automated multicomponent identification at complex material interfaces](https://mlsanity.com/paper/2609.14348/) (2026)
- [On Making SE(2)-Invariant Networks Optimal](https://mlsanity.com/paper/2610.17424/) (2026)

## Abstract (arXiv)

In recent years, convolutional neural network has shown good performance in many image processing and computer vision tasks. However, a standard CNN model is not invariant to image rotations. In fact, even slight rotation of an input image will seriously degrade its performance. This shortcoming precludes the use of CNN in some practical scenarios. Thus, in this paper, we focus on designing convolutional layer with good rotation invariance. Specifically, based on a simple rotation-invariant coordinate system, we propose a new convolutional operation, called Rotation-Invariant Coordinate Convolution (RIC-C). Without additional trainable parameters and data augmentation, RIC-C is naturally invariant to arbitrary rotations around the input center. Furthermore, we find the connection between RIC-C and deformable convolution, and propose a simple but efficient approach to implement RIC-C using Pytorch. By replacing all standard convolutional layers in a CNN with the corresponding RIC-C, a RIC-CNN can be derived. Using MNIST dataset, we first evaluate the rotation invariance of RIC-CNN and compare its performance with most of existing rotation-invariant CNN models. It can be observed that RIC-CNN achieves the state-of-the-art classification on the rotated test dataset of MNIST. Then, we deploy RIC-C to VGG, ResNet and DenseNet, and conduct the classification experiments on two real image datasets. Also, a shallow CNN and the corresponding RIC-CNN are trained to extract image patch descriptors, and we compare their performance in patch verification. These experimental results again show that RIC-C can be easily used as drop in replacement for standard convolutions, and greatly enhances the rotation invariance of CNN models designed for different applications.