# Scale-Equivariant Deep Learning for 3D Data

arXiv: https://arxiv.org/abs/2304.05864 · HTML: https://mlsanity.com/paper/2304.05864/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 35 references that are in the map.
- First posted on arXiv 12 Apr 2023.

## Builds on (most cited first, 27 of 27)

- [V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation](https://mlsanity.com/paper/1606.04797/) (2016)
- [Rich feature hierarchies for accurate object detection and semantic segmentation](https://mlsanity.com/paper/1311.2524/) (2013)
- [3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation](https://mlsanity.com/paper/1606.06650/) (2016)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [Identifying the Best Machine Learning Algorithms for Brain Tumor Segmentation, Progression Assessment, and Overall Survival Prediction in the BRATS Challenge](https://mlsanity.com/paper/1811.02629/) (2018)
- [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)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Drop an Octave: Reducing Spatial Redundancy in Convolutional Neural Networks with Octave Convolution](https://mlsanity.com/paper/1904.05049/) (2019)
- [Deep Scale-spaces: Equivariance Over Scale](https://mlsanity.com/paper/1905.11697/) (2019)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [nnU-Net for Brain Tumor Segmentation](https://mlsanity.com/paper/2011.00848/) (2020)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Scale-Invariant Convolutional Neural Networks](https://mlsanity.com/paper/1411.6369/) (2014)
- [DISCO: accurate Discrete Scale Convolutions](https://mlsanity.com/paper/2106.02733/) (2021)
- [Locally Scale-Invariant Convolutional Neural Networks](https://mlsanity.com/paper/1412.5104/) (2014)
- [Scale-covariant and scale-invariant Gaussian derivative networks](https://mlsanity.com/paper/2011.14759/) (2020)
- [Scale Steerable Filters for Locally Scale-Invariant Convolutional Neural Networks](https://mlsanity.com/paper/1906.03861/) (2019)
- [Scale equivariance in CNNs with vector fields](https://mlsanity.com/paper/1807.11783/) (2018)
- [Rotation-Equivariant Deep Learning for Diffusion MRI](https://mlsanity.com/paper/2102.06942/) (2021)
- [Scaling-Translation-Equivariant Networks with Decomposed Convolutional Filters](https://mlsanity.com/paper/1909.11193/) (2019)
- [Scale-invariant scale-channel networks: Deep networks that generalise to previously unseen scales](https://mlsanity.com/paper/2106.06418/) (2021)
- [Scale-Equivariant UNet for Histopathology Image Segmentation](https://mlsanity.com/paper/2304.04595/) (2023)
- [Scale-Equivariant Unrolled Neural Networks for Data-Efficient Accelerated MRI Reconstruction](https://mlsanity.com/paper/2204.10436/) (2022)

## Built on by (most cited first, 6 of 6)

- [Unified theory for joint covariance properties under geometric image transformations for spatio-temporal receptive fields according to the generalized Gaussian derivative model for visual receptive fields](https://mlsanity.com/paper/2311.10543/) (2023)
- [Scale generalisation properties of extended scale-covariant and scale-invariant Gaussian derivative networks on image datasets with spatial scaling variations](https://mlsanity.com/paper/2409.11140/) (2024)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [EquiFusion: Kinematics-Agnostic Human Motion Prediction via Equivariant Latent Diffusion](https://mlsanity.com/paper/2607.10984/) (2026)
- [Hybrid Lie semi-group and cascade structures for the generalized Gaussian derivative model for visual receptive fields](https://mlsanity.com/paper/2509.15748/) (2025)
- [Scale-invariant Gaussian derivative residual networks](https://mlsanity.com/paper/2603.02843/) (2026)

## Abstract (arXiv)

The ability of convolutional neural networks (CNNs) to recognize objects regardless of their position in the image is due to the translation-equivariance of the convolutional operation. Group-equivariant CNNs transfer this equivariance to other transformations of the input. Dealing appropriately with objects and object parts of different scale is challenging, and scale can vary for multiple reasons such as the underlying object size or the resolution of the imaging modality. In this paper, we propose a scale-equivariant convolutional network layer for three-dimensional data that guarantees scale-equivariance in 3D CNNs. Scale-equivariance lifts the burden of having to learn each possible scale separately, allowing the neural network to focus on higher-level learning goals, which leads to better results and better data-efficiency. We provide an overview of the theoretical foundations and scientific work on scale-equivariant neural networks in the two-dimensional domain. We then transfer the concepts from 2D to the three-dimensional space and create a scale-equivariant convolutional layer for 3D data. Using the proposed scale-equivariant layer, we create a scale-equivariant U-Net for medical image segmentation and compare it with a non-scale-equivariant baseline method. Our experiments demonstrate the effectiveness of the proposed method in achieving scale-equivariance for 3D medical image analysis. We publish our code at https://github.com/wimmerth/scale-equivariant-3d-convnet for further research and application.