# Equivariant Spherical Deconvolution: Learning Sparse Orientation Distribution Functions from Spherical Data

arXiv: https://arxiv.org/abs/2102.09462 · HTML: https://mlsanity.com/paper/2102.09462/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 22 references that are in the map.
- First posted on arXiv 17 Feb 2021.

## Builds on (most cited first, 5 of 5)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications](https://mlsanity.com/paper/1810.12186/) (2018)
- [A machine learning-based method for estimating the number and orientations of major fascicles in diffusion-weighted magnetic resonance imaging](https://mlsanity.com/paper/2006.11117/) (2020)
- [Enhancing Fiber Orientation Distributions using convolutional Neural Networks](https://mlsanity.com/paper/2008.05409/) (2020)

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

- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Diffusion MRI with Machine Learning](https://mlsanity.com/paper/2402.00019/) (2024)
- [How can spherical CNNs benefit ML-based diffusion MRI parameter estimation?](https://mlsanity.com/paper/2207.00572/) (2022)
- [Recovering high-quality FODs from a reduced number of diffusion-weighted images using a model-driven deep learning architecture](https://mlsanity.com/paper/2307.15273/) (2023)
- [Spherical convolutional neural networks can improve brain microstructure estimation from diffusion MRI data](https://mlsanity.com/paper/2211.09887/) (2022)
- [E(3) × SO(3)-Equivariant Networks for Spherical Deconvolution in Diffusion MRI](https://mlsanity.com/paper/2304.06103/) (2023)
- [Equivariant spatio-hemispherical networks for diffusion MRI deconvolution](https://mlsanity.com/paper/2411.11819/) (2024)
- [Prior-Guided Implicit Neural Representations for Single-Subject Diffusion MRI Super-Resolution](https://mlsanity.com/paper/2609.00981/) (2026)
- [GORDA: Graph-based ORientation Distribution Analysis of SLI scatterometry Patterns of Nerve Fibres](https://mlsanity.com/paper/2204.05776/) (2022)
- [Neural Operators for Biomedical Spherical Heterogeneity](https://mlsanity.com/paper/2601.03561/) (2026)

## Abstract (arXiv)

We present a rotation-equivariant unsupervised learning framework for the sparse deconvolution of non-negative scalar fields defined on the unit sphere. Spherical signals with multiple peaks naturally arise in Diffusion MRI (dMRI), where each voxel consists of one or more signal sources corresponding to anisotropic tissue structure such as white matter. Due to spatial and spectral partial voluming, clinically-feasible dMRI struggles to resolve crossing-fiber white matter configurations, leading to extensive development in spherical deconvolution methodology to recover underlying fiber directions. However, these methods are typically linear and struggle with small crossing-angles and partial volume fraction estimation. In this work, we improve on current methodologies by nonlinearly estimating fiber structures via unsupervised spherical convolutional networks with guaranteed equivariance to spherical rotation. Experimentally, we first validate our proposition via extensive single and multi-shell synthetic benchmarks demonstrating competitive performance against common baselines. We then show improved downstream performance on fiber tractography measures on the Tractometer benchmark dataset. Finally, we show downstream improvements in terms of tractography and partial volume estimation on a multi-shell dataset of human subjects.