# SE(3)-Equivariant and Noise-Invariant 3D Rigid Motion Tracking in Brain MRI

arXiv: https://arxiv.org/abs/2312.13534 · HTML: https://mlsanity.com/paper/2312.13534/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 78 references that are in the map.
- First posted on arXiv 21 Dec 2023.

## Builds on (most cited first, 30 of 33)

- [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)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Multimodal Unsupervised Image-to-Image Translation](https://mlsanity.com/paper/1804.04732/) (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)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [SynthSeg: Segmentation of brain MRI scans of any contrast and resolution without retraining](https://mlsanity.com/paper/2107.09559/) (2021)
- [TorchIO: A Python library for efficient loading, preprocessing, augmentation and patch-based sampling of medical images in deep learning](https://mlsanity.com/paper/2003.04696/) (2020)
- [A Deep Learning Framework for Unsupervised Affine and Deformable Image Registration](https://mlsanity.com/paper/1809.06130/) (2018)
- [SynthMorph: learning contrast-invariant registration without acquired images](https://mlsanity.com/paper/2004.10282/) (2020)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [Roto-Translation Covariant Convolutional Networks for Medical Image Analysis](https://mlsanity.com/paper/1804.03393/) (2018)
- [Residual Pathway Priors for Soft Equivariance Constraints](https://mlsanity.com/paper/2112.01388/) (2021)
- [Disentangled Representation Learning in Cardiac Image Analysis](https://mlsanity.com/paper/1903.09467/) (2019)
- [CubeNet: Equivariance to 3D Rotation and Translation](https://mlsanity.com/paper/1804.04458/) (2018)
- [The Lie Derivative for Measuring Learned Equivariance](https://mlsanity.com/paper/2210.02984/) (2022)
- [Robust machine learning segmentation for large-scale analysis of heterogeneous clinical brain MRI datasets](https://mlsanity.com/paper/2209.02032/) (2022)
- [Learning Disentangled Representations in the Imaging Domain](https://mlsanity.com/paper/2108.12043/) (2021)
- [Real-time Deep Pose Estimation with Geodesic Loss for Image-to-Template Rigid Registration](https://mlsanity.com/paper/1803.05982/) (2018)
- [3D G-CNNs for Pulmonary Nodule Detection](https://mlsanity.com/paper/1804.04656/) (2018)
- [Roto-Translation Equivariant Convolutional Networks: Application to Histopathology Image Analysis](https://mlsanity.com/paper/2002.08725/) (2020)
- [A Robust and Interpretable Deep Learning Framework for Multi-modal Registration via Keypoints](https://mlsanity.com/paper/2304.09941/) (2023)
- [An Artificial Agent for Robust Image Registration](https://mlsanity.com/paper/1611.10336/) (2016)
- [Equivariant Filters for Efficient Tracking in 3D Imaging](https://mlsanity.com/paper/2103.10255/) (2021)
- [Intertwiners between Induced Representations (with Applications to the Theory of Equivariant Neural Networks)](https://mlsanity.com/paper/1803.10743/) (2018)
- [AIRNet: Self-Supervised Affine Registration for 3D Medical Images using Neural Networks](https://mlsanity.com/paper/1810.02583/) (2018)

## Built on by (most cited first, 10 of 10)

- [Segment as You Wish -- Free-Form Language-Based Segmentation for Medical Images](https://mlsanity.com/paper/2410.12831/) (2024)
- [CABLD: Contrast-Agnostic Brain Landmark Detection with Consistency-Based Regularization](https://mlsanity.com/paper/2411.17845/) (2024)
- [Spatial regularisation for improved accuracy and interpretability in keypoint-based registration](https://mlsanity.com/paper/2503.04499/) (2025)
- [Fetuses Made Simple: Modeling and Tracking of Fetal Shape and Pose](https://mlsanity.com/paper/2506.17858/) (2025)
- [Intraoperative 2D/3D Registration via Spherical Similarity Learning and Differentiable Levenberg-Marquardt Optimization](https://mlsanity.com/paper/2509.06890/) (2025)
- [A Unified Deep Learning Framework for Motion Correction in Medical Imaging](https://mlsanity.com/paper/2409.14204/) (2024)
- [Moment kernels: a simple and scalable approach for equivariance to rotations and reflections in deep convolutional networks](https://mlsanity.com/paper/2505.21736/) (2025)
- [Equivariant symmetry-aware head pose estimation for fetal MRI](https://mlsanity.com/paper/2512.04890/) (2025)
- [Rotation Equivariant Convolutions in Deformable Registration of Brain MRI](https://mlsanity.com/paper/2604.08034/) (2026)
- [Patient-Agnostic Synthetic Pretraining for Efficient Patient-Specific Intraoperative 2D/3D Registration](https://mlsanity.com/paper/2607.23343/) (2026)

## Abstract (arXiv)

Rigid motion tracking is paramount in many medical imaging applications where movements need to be detected, corrected, or accounted for. Modern strategies rely on convolutional neural networks (CNN) and pose this problem as rigid registration. Yet, CNNs do not exploit natural symmetries in this task, as they are equivariant to translations (their outputs shift with their inputs) but not to rotations. Here we propose EquiTrack, the first method that uses recent steerable SE(3)-equivariant CNNs (E-CNN) for motion tracking. While steerable E-CNNs can extract corresponding features across different poses, testing them on noisy medical images reveals that they do not have enough learning capacity to learn noise invariance. Thus, we introduce a hybrid architecture that pairs a denoiser with an E-CNN to decouple the processing of anatomically irrelevant intensity features from the extraction of equivariant spatial features. Rigid transforms are then estimated in closed-form. EquiTrack outperforms state-of-the-art learning and optimisation methods for motion tracking in adult brain MRI and fetal MRI time series. Our code is available at https://github.com/BBillot/EquiTrack.