# E(3) × SO(3)-Equivariant Networks for Spherical Deconvolution in Diffusion MRI

arXiv: https://arxiv.org/abs/2304.06103 · HTML: https://mlsanity.com/paper/2304.06103/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 50 references that are in the map.
- First posted on arXiv 12 Apr 2023.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications](https://mlsanity.com/paper/1810.12186/) (2018)
- [The Lie Derivative for Measuring Learned Equivariance](https://mlsanity.com/paper/2210.02984/) (2022)
- [Scalable and Equivariant Spherical CNNs by Discrete-Continuous (DISCO) Convolutions](https://mlsanity.com/paper/2209.13603/) (2022)
- [Rotation-Equivariant Deep Learning for Diffusion MRI](https://mlsanity.com/paper/2102.06942/) (2021)
- [VolterraNet: A higher order convolutional network with group equivariance for homogeneous manifolds](https://mlsanity.com/paper/2106.15301/) (2021)
- [Equivariant Spherical Deconvolution: Learning Sparse Orientation Distribution Functions from Spherical Data](https://mlsanity.com/paper/2102.09462/) (2021)
- [How can spherical CNNs benefit ML-based diffusion MRI parameter estimation?](https://mlsanity.com/paper/2207.00572/) (2022)
- [Deep Learning Estimation of Multi-Tissue Constrained Spherical Deconvolution with Limited Single Shell DW-MRI](https://mlsanity.com/paper/2002.08820/) (2020)

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

- [Diffusion MRI with Machine Learning](https://mlsanity.com/paper/2402.00019/) (2024)

## Abstract (arXiv)

We present Roto-Translation Equivariant Spherical Deconvolution (RT-ESD), an E(3)× SO(3) equivariant framework for sparse deconvolution of volumes where each voxel contains a spherical signal. Such 6D data naturally arises in diffusion MRI (dMRI), a medical imaging modality widely used to measure microstructure and structural connectivity. As each dMRI voxel is typically a mixture of various overlapping structures, there is a need for blind deconvolution to recover crossing anatomical structures such as white matter tracts. Existing dMRI work takes either an iterative or deep learning approach to sparse spherical deconvolution, yet it typically does not account for relationships between neighboring measurements. This work constructs equivariant deep learning layers which respect to symmetries of spatial rotations, reflections, and translations, alongside the symmetries of voxelwise spherical rotations. As a result, RT-ESD improves on previous work across several tasks including fiber recovery on the DiSCo dataset, deconvolution-derived partial volume estimation on real-world textitin vivo human brain dMRI, and improved downstream reconstruction of fiber tractograms on the Tractometer dataset. Our implementation is available at https://github.com/AxelElaldi/e3so3conv