# Equivariant Filters for Efficient Tracking in 3D Imaging

arXiv: https://arxiv.org/abs/2103.10255 · HTML: https://mlsanity.com/paper/2103.10255/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 22 references that are in the map.
- First posted on arXiv 18 Mar 2021.

## Builds on (most cited first, 9 of 9)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [On the Continuity of Rotation Representations in Neural Networks](https://mlsanity.com/paper/1812.07035/) (2018)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [A Deep Learning Framework for Unsupervised Affine and Deformable Image Registration](https://mlsanity.com/paper/1809.06130/) (2018)
- [TVM: An Automated End-to-End Optimizing Compiler for Deep Learning](https://mlsanity.com/paper/1802.04799/) (2018)
- [Real-time Deep Pose Estimation with Geodesic Loss for Image-to-Template Rigid Registration](https://mlsanity.com/paper/1803.05982/) (2018)
- [Finding Symmetry Breaking Order Parameters with Euclidean Neural Networks](https://mlsanity.com/paper/2007.02005/) (2020)
- [Slice-to-volume medical image registration: a survey](https://mlsanity.com/paper/1702.01636/) (2017)
- [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, 17 of 17)

- [A Robust and Interpretable Deep Learning Framework for Multi-modal Registration via Keypoints](https://mlsanity.com/paper/2304.09941/) (2023)
- [Learning Rotation-Equivariant Features for Visual Correspondence](https://mlsanity.com/paper/2303.15472/) (2023)
- [Anatomy-aware and acquisition-agnostic joint registration with SynthMorph](https://mlsanity.com/paper/2301.11329/) (2023)
- [SE(3)-Equivariant and Noise-Invariant 3D Rigid Motion Tracking in Brain MRI](https://mlsanity.com/paper/2312.13534/) (2023)
- [Deep Implicit Optimization enables Robust Learnable Features for Deformable Image Registration](https://mlsanity.com/paper/2406.07361/) (2024)
- [Equivariant Neural Tangent Kernels](https://mlsanity.com/paper/2406.06504/) (2024)
- [Rigid Single-Slice-in-Volume registration via rotation-equivariant 2D/3D feature matching](https://mlsanity.com/paper/2410.18683/) (2024)
- [Learning accurate rigid registration for longitudinal brain MRI from synthetic data](https://mlsanity.com/paper/2501.13010/) (2025)
- [SpaER: Learning Spatio-temporal Equivariant Representations for Fetal Brain Motion Tracking](https://mlsanity.com/paper/2407.20198/) (2024)
- [Bayesian Unsupervised Disentanglement of Anatomy and Geometry for Deep Groupwise Image Registration](https://mlsanity.com/paper/2401.02141/) (2024)
- [CARL: A Framework for Equivariant Image Registration](https://mlsanity.com/paper/2405.16738/) (2024)
- [Spatial regularisation for improved accuracy and interpretability in keypoint-based registration](https://mlsanity.com/paper/2503.04499/) (2025)
- [Intraoperative 2D/3D Registration via Spherical Similarity Learning and Differentiable Levenberg-Marquardt Optimization](https://mlsanity.com/paper/2509.06890/) (2025)
- [Registering Image Volumes using 3D SIFT and Discrete SP-Symmetry](https://mlsanity.com/paper/2205.15456/) (2022)
- [A Unified Deep Learning Framework for Motion Correction in Medical Imaging](https://mlsanity.com/paper/2409.14204/) (2024)
- [Equivariant symmetry-aware head pose estimation for fetal MRI](https://mlsanity.com/paper/2512.04890/) (2025)
- [Patient-Agnostic Synthetic Pretraining for Efficient Patient-Specific Intraoperative 2D/3D Registration](https://mlsanity.com/paper/2607.23343/) (2026)

## Abstract (arXiv)

We demonstrate an object tracking method for 3D images with fixed computational cost and state-of-the-art performance. Previous methods predicted transformation parameters from convolutional layers. We instead propose an architecture that does not include either flattening of convolutional features or fully connected layers, but instead relies on equivariant filters to preserve transformations between inputs and outputs (e.g. rot./trans. of inputs rotate/translate outputs). The transformation is then derived in closed form from the outputs of the filters. This method is useful for applications requiring low latency, such as real-time tracking. We demonstrate our model on synthetically augmented adult brain MRI, as well as fetal brain MRI, which is the intended use-case.