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Equivariant spatio-hemispherical networks for diffusion MRI deconvolution
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Proposes SHD, an E(3)×SO(3)-equivariant spatio-hemispherical U-Net for dMRI fODF deconvolution that halves the spherical graph to a hemisphere, uses dense matmul and precomputed Chebyshev polynomials, and adds TV spatial regularization (SHD-TV), yielding faster runtime and state-of-the-art fiber detection.</tl;dr>
Problem
Equivariant spatio-spherical networks for dMRI fiber deconvolution (fODF recovery) are too computationally expensive (RT-ESD takes about a day per subject) for clinical or large-scale use, and existing learned methods either ignore spatial correlation or rely on supervised data.
Approach
Exploit antipodal symmetry of fODFs to reduce the spherical graph Laplacian to a hemispherical one (L+ = L(p,q)+L(p,-q)), replace sparse with dense matmul, precompute fixed Chebyshev polynomials, and add an explicit spatial total-variation regularizer within an unsupervised reconstruction loss.
Key results
- SHD-TV achieves the best DiSCo high-resolution F1 of 0.65 and angular error of 17.5 degrees, versus 0.60 and 20.4 for RT-ESD.
- The hemispherical U-Net runs about 2.5x faster than RT-ESD and 3.5x faster than PONITA, using up to 4x and 20x less memory respectively.
- On Tractometer, SHD-TV significantly outperforms all baselines in fODF estimation and tractography.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add total-variation spatial regularization to SHD (SHD-TV) (ablation) | SHD · DiSCo high-angular | PR AUC: 0.64 -> 0.70 | Helped |
| Add total-variation spatial regularization to SHD (SHD-TV) (ablation) | SHD · DiSCo low-angular | PR AUC: 0.61 -> 0.62 | Helped |
| Add total-variation spatial regularization to SHD (SHD-TV) (ablation) | SHD · DiSCo high-angular | Angle @F1: 20.0 ± 0.1 -> 17.5 ± 0.2 | Helped |
| Use SHD instead of full-sphere RT-ESD convolutions | RT-ESD · DiSCo low-angular | PR AUC: 0.60 -> 0.61 | Helped |
| Use spatial information in the network input | Deconvolution network · DiSCo high-angular | fiber detection performance | No clear effect |
| Use spatially informed models (SHD and SHD-TV) rather than voxel-wise methods | Deconvolution network · DiSCo low-angular | fiber detection performance | Helped |
| Use hemispherical convolutions instead of spherical convolutions (ablation) | SHD convolution · High-angular-resolution convolution efficiency | runtime | Helped |
| Precompute Laplacian polynomials (ablation) | SHD convolution · High-angular-resolution convolution efficiency | runtime | Helped |
| Use dense matrix multiplication instead of sparse matrix multiplication (ablation) | SHD convolution · Low-angular-resolution convolution efficiency | runtime | Helped |
Surprising findings
- Non-equivariant CNN and CNN-TV outperform most unregularized equivariant methods on Tractometer, which the authors attribute to overfitting capacity when generalization is not needed.
Limitations
- Assumes antipodal symmetry of fODFs and dMRI, which may need relaxing for other applications
- Undersampled reconstruction risks missing fibers or hallucinating fibers
- Not yet validated on clinical challenges such as motion, artifacts, or abnormal structures; glioma result is preliminary
Experimental setup
Datasets: DiSCo HCP Young Adult Tractometer
Benchmarks: DiSCo HCP Young Adult Tractometer
Metrics: PR AUC F1 score angular error false positive rate false negative rate valid bundles valid streamlines overlap overreach runtime
Baselines: CSD RUMBA RUMBA-TV MLP CNN ESD RT-ESD PONITA
Builds on: RT-ESD DeepSphere ESD CSD RUMBA
Closest papers by meaning
- TractoTransformer: Diffusion MRI Streamline Tractography using CNN and Transformer Networks (NeurIPS 2025 · similarity 0.88)
- On the Fourier analysis in the SO(3) space : the EquiLoPO Network (ICLR 2025 · similarity 0.86)
- DiffATS: Diffusion in Aligned Tensor Space (NeurIPS 2026 · similarity 0.85)
- A Steerable Deep Network for Model-Free Diffusion MRI Registration (NeurIPS 2026 · similarity 0.85)
- MAGNET: Manifold-Aware Graph Diffusion Network for Connectome Generation (NeurIPS 2026 · similarity 0.84)
- Equivariant Neural Tangent Kernels (ICML 2025 · similarity 0.84)
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