Title: Equivariant spatio-hemispherical networks for diffusion MRI deconvolution
Venue: NeurIPS 2024
Accepted: yes
Keywords: Geometric Deep Learning

Diffusion MRI

Spherical Networks

Biomedical Image Analysis
Abstract: Each voxel in a diffusion MRI (dMRI) image contains a spherical signal corresponding to the direction and strength of water diffusion in the brain. This paper advances the analysis of such spatio-spherical data by developing convolutional network layers that are equivariant to the $\mathbf{E(3) \times SO(3)}$ group and account for the physical symmetries of dMRI including rotations, translations, and reflections of space alongside voxel-wise rotations. Further, neuronal fibers are typically antipodally symmetric, a fact we leverage to construct highly efficient spatio-*hemispherical* graph convolutions to accelerate the analysis of high-dimensional dMRI data. In the context of sparse spherical fiber deconvolution to recover white matter microstructure, our proposed equivariant network layers yield substantial performance and efficiency gains, leading to better and more practical resolution of crossing neuronal fibers and fiber tractography. These gains are experimentally consistent across both simulation and in vivo human datasets.

Equivariant spatio-hemispherical networks for diffusion MRI deconvolution

Abstract

Each voxel in a diffusion MRI (dMRI) image contains a spherical signal corresponding to the direction and strength of water diffusion in the brain. This paper advances the analysis of such spatio-spherical data by developing convolutional network layers that are equivariant to the E ( 3 ) × SO ( 3 ) group and account for the physical symmetries of dMRI including rotations, translations, and reflections of space alongside voxel-wise rotations. Further, neuronal fibers are typically antipodally symmetric, a fact we leverage to construct highly efficient spatiohemispherical graph convolutions to accelerate the analysis of high-dimensional dMRI data. In the context of sparse spherical fiber deconvolution to recover white matter microstructure, our proposed equivariant network layers yield substantial performance and efficiency gains, leading to better and more practical resolution of crossing neuronal fibers and fiber tractography. These gains are experimentally consistent across both simulation and in vivo human datasets.

1 Introduction

Instead of scalar intensities, each voxel of a diffusion MR image (dMRI) contains spatio-angular measurements of local water diffusion [64]. As in Fig. 1, this yields spatio-spherical images living on R 3 ×S 2 that are used to map neuronal organization in vivo [7, 44] alongside several other biomedical use cases [35, 37, 56, 65, 71]. Despite its potential, deriving neuronal fiber pathways using dMRI is hampered by significant partial voluming in both spatial and spherical domains due to limited clinical scanner resolutions and low SNR [1]. This paper presents a state-of-the-art dMRI deconvolution method to recover neuronal pathways in practical timeframes by developing highly efficient equivariant neural networks that account for dMRI's spatio-spherical data geometry and the voxel-level antipodal symmetry of neuronal fibers, all while demonstrating robustness to clinical dMRI resolutions.

Need for deconvolution. The diffusivity at a voxel reflects the underlying local tissue microstructure [7, 8, 44]. As multiple neuronal fibers can cross within a given resolution-limited voxel, recovering these fibers necessitates solving a blind fiber deconvolution problem at each voxel. Several voxel-level fiber models have been proposed [5, 45, 67, 69, 73, 79] and this work focuses on the widely used fiber Orientation Distribution Function (fODF) model, which represents the fiber configuration at a voxel as a sparse non-negative spherical function [66]. fODFs are the first step of common dMRI applications such as fiber tractography [37]. However, due to noise, subject motion, and clinically viable sampling resolutions (e.g., ≤ 30 spherical samples per voxel), fODF recovery is highly ill-posed and requires significant regularization.

Iterative solutions. Constrained spherical deconvolution (CSD) [66] is the workhorse algorithm for dMRI deconvolution, solving the per-voxel fODF deconvolution problem iteratively, subject to non-negativity constraints. Several extensions of CSD further regularize fODFs to be sparse and/or spatially smooth [11, 12, 29, 57, 59, 74]. Further, as fODFs can be assumed to be antipodally symmetric [40, 67], many iterative algorithms optimize only over the hemisphere to accelerate

Neel Dey MIT CSAIL

Diffusion MRI

T1 MRI

Gradient1

Gradient25

Gradient288

Spatio-SphericalSignal

per-voxel optimization. For example, CSD only optimizes for the even-order spherical harmonics of the fODF and RUMBA [11] uses approximately hemispherical sampling to represent the fODF. Despite significant progress using iterative solutions, these methods still require high angular sampling resolutions that are not clinically feasible to resolve crossing fibers within a voxel.

Deep deconvolution networks. Recent deep learning frameworks for dMRI deconvolution incorporate supervision and/or additional inductive biases to improve results. Supervised approaches train U-Nets to regress the solutions of iterative models or regress ground-truth fODFs estimated from ex vivo animal histology. These methods are consequently upper-bounded by the quality of iterative solutions and the generalizability limitations of small animal datasets. Further extensions incorporate the underlying spherical geometry of the per-voxel estimation by using SO ( 3 ) -equivariant network layers but retain the supervised strategy of previous models [62]. ESD [19] instead proposes to use SO ( 3 ) -equivariant network layers and unsupervised and regularized reconstruction-based deconvolution losses to surpass supervised solutions.

However, the above methods operate entirely at the voxel level and do not model the strong correlation between neighboring fODFs. To this end, RT-ESD [20] proposed E ( 3 ) × SO ( 3 ) -equivariant network layers for dMRI that are equivariant to spatial rotations, translations, and reflections alongside voxel-wise spherical rotations to achieve high-quality deconvolution due to the joint spatio-spherical modeling. However, its high computational requirements limit its use in clinical or large-scale deployment. For instance, while subject-specific iterative fits with CSD require 3-5 minutes for the whole brain, RT-ESD requires about a day on an A100 GPU to converge when trained on a single subject.

Contributions. We present an efficient equivariant neural network framework for dMRI deconvolution that respects the data geometry of dMRI while ensuring computational practicality. Our spatio-hemispherical deconvolution (SHD) method addresses key redundancies and weaknesses of previous deep networks for dMRI deconvolution in three ways: (1) As fODFs are approximately antipodally symmetric at clinical resolutions, we replace RT-ESD's full spherical sampling with hemispherical sampling and find substantial efficiency gains by reducing the graph Laplacian of the voxel-wise SO(3) convolutions used in RT-ESD; (2) We then exploit the dense structure of dMRI data to further introduce optimized implementations and pre-computations, achieving a cumulative 65% reduction in processing time for an E(3) × SO(3)-equivariant U-Net; (3) Finally, we use explicit smoothness-promoting spatial regularization, leading to further improved fODF recovery.

Experimentally, we achieve state-of-the-art deconvolution results on two widely used simulated dMRI benchmarks with known ground truth. On real in vivo human dMRI, our method yields more spatially coherent fODF fields and higher robustness to changes in resolution from research-grade to clinical standards of single-shell low-angular protocols. Lastly, as the achieved efficiency gains enable training on a large set of human datasets, we can now train a single network for amortized inference on new human dMRI, as opposed to the subject-specific optimization of RT-ESD. Our code is available at https://github.com/AxelElaldi/fast-equivariant-deconv .

2 Related work

Deep learning for dMRI. Learned networks operating on dMRI have been extensively used to improve fiber tractography [46, 63], super-resolve diffusion signals or fODFs [48, 58, 78], regress

A.1) CSD

Multi-crossing area

A.1) CSD

3-Shell High-Angular Resolution

1-Shell Low-Angular Resolution

Cerebellum

B.1) SHD-TV (ours)

B.1) SHD-TV (ours)

Multi-crossing area

3-Shell High-Angular Resolution

1-Shell Low-Angular Resolution

Cerebellum

microstructral indices in undersampled settings [13, 14, 27, 30, 31, 61], denoise artifacts [22, 23, 76], segment lesions [50], and more. Of these, most employ supervision via training on high-angular resolution research datasets to generalize to low-resolution clinical datasets. Our work is most related to networks that directly operate on R 3 ×S 2 images either through spherical networks applied voxelwise or approximate parameterizations of the spatio-spherical space [6, 9, 10, 20, 19, 47, 50, 62].

For spherical data, SO ( 3 ) -equivariant convolutions are often achieved via spherical harmonicsparameterized convolutions [16, 21, 42] or isotropic convolutions on spherical graphs [53]. These models have been extended to voxel-wise fODF estimation [19, 62] but do not use the spatial correlation between fODFs. [6, 10] develop convolutions for dMRI classification and super-resolution that exhibit voxel-wise SO ( 3 ) -equivariance and incorporate manifold-valued spatial averaging. To model spatio-spherical dependence explicitly, SE ( 3 ) -equivariance for improved dMRI segmentation has been achieved via tensor field networks [50] and separable kernels [9, 47] at the cost of high-memory usage. In contrast, RT-ESD [20] uses isotropic spatio-spherical kernels and focuses on deconvolution with equivariance to both joint and independent voxel and grid rotations. More recently, PONITA [9] proposes efficient SE ( 3 ) -equivariant convolutions for spatio-spherical molecular graphs, but as yet, does not scale to high-angular resolutions needed for dMRI deconvolution.

Spherical deconvolution methods. Constrained spherical deconvolution (CSD) [38, 66] and its sparse [12] and/or spatially regularized extensions [11, 29, 57, 59, 74] are established but rely on lengthy and data-dependent iterative optimization and struggle to resolve small-angle crossing fibers in the low-resolution setting (see Fig. 2). More recently, several trainable models have been proposed for direct fODF estimation including pattern matching methods [24] that use a dictionary to match diffusion signal to known tissue microstructure and deep neural networks [19, 20, 39, 43, 46, 51, 52, 61, 62, 78] that learn a mapping from diffusion signals to fODFs. Trainable models have the advantage of potentially decreasing the need for high-angular resolution dMRI input. ESD [19] further introduced an unsupervised learning framework using only a diffusion signal reconstruction loss and additional fODF regularization, removing the need for ground-truth fODFs for training.

Spatially informed deconvolution. Traditionally, spherical deconvolution is optimized voxel-wise. However, the underlying tissue microstructure has long-range spatial correlations. Iterative methods account for this through spatial regularization such as total-variation [11] or fiber continuity [29, 57, 59, 74], at the cost of increased optimization complexity. Current neural network frameworks

have overlooked explicit fODF spatial regularization and rely solely on high-quality fODF training data and implicit spatial regularization by mean of spatial-weight sharing, either employing grid-wise 3D convolution [20, 46, 78], or channel-wise concatenation of neighboring voxels [19, 61, 62].

3 Methods

3.1 Background

dMRI deconvolution. A dMRI image is a spatio-spherical function S : R 3 ×S 2 → R B with B features, called shells in the dMRI literature. The fODF model describes the tissue microstructure as a non-negative spatio-spherical function F , providing information on the local tissue composition and orientations, and links the dMRI and the fODFs S = C ( F ) . dMRI deconvolution is interested in recovering the fODF from the dMRI by minimizing a constrained reconstruction loss F ∗ = argmin F ≥ 0 || S -C ( F ) || 2 2 . We focus our study on learnable spatio-spherical deconvolution operators trained to reconstruct high-angular resolution fODFs from sparse dMRI measurements N θ ( S ) = F .

Spatio-spherical convolutions. E ( 3 ) × SO ( 3 ) is the group of independent E ( 3 ) and SO ( 3 ) transformations, where E ( 3 ) acts on the spatial grid R 3 and SO ( 3 ) acts on the voxel sphere S 2 . Let ψ θ : R 3 × S 2 → R be a learnable filter, such that convolving f and ψ θ yields f out ( T, R ) = ∫ y ∈ R 3 ∫ p ∈S 2 ψ θ ( T -1 y, R -1 p ) f ( y, p ) dpdy where ( T, R ) ∈ E ( 3 ) × SO ( 3 ) . RTESD [20] implements these convolutions by only considering isotropic filters to limit computational complexity. In addition, the filter ψ θ is expressed as a separable kernel ψ θ ( z 1 , z 2 ) = ϕ θ 1 ( z 1 ) ϕ θ 2 ( z 2 ) , turning the convolution into a two-step spherical-spatial convolution:

We build on RT-ESD [20], presented in Fig. 12, which uses DeepSphere [53]'s graph filtering based SO ( 3 ) -equivariant spherical convolution. At every spatial location y ∈ R 3 , f ( y, . ) is discretized on a set of spherical vertices V = { q i ∈ S 2 } i ∈ [1 ,.,N V ] , f ( y ) ∈ R N V . The graph G = ( V , A ) and its normalized Laplacian L are constructed from the set V , where A ∈ R N V × N V is the spherical adjacency matrix. The voxel-wise spherical filtering output is then used to construct a spatio-spherical graph ˆ f ( y ) = [ T 0 ( L ) f ( y ) , ..., T K -1 ( L ) f ( y )] ∈ R N V × K with K features, where K is the polynomial degree of the filtering and T k ( L ) is the Chebyshev polynomial of degree k . A learnable isotropic spatial filter α k : R + → R is then defined for each spherical filtered map ˆ f k , resulting in the overall E ( 3 ) × SO ( 3 ) -convolutional operation expressed as:

where P ( x ) is a spatial neighborhood around the position x ∈ R 3 . Spherical SO ( 3 ) -equivariance is achieved by using a uniform and symmetric V sampling such as the HEALPix [32] grid, providing a uniform coverage of the spherical domain at different levels of resolution, and graph weight p, q ∈ V , A ( p, q ) = e -|| p -q || /σ with σ 's value selected to minimize empirical equivariance error.

3.2 Efficient dMRI spatio-hemispherical convolutions

Spatio-Hemispherical Equivariant convolution. fODFs and dMRIs are approximately antipodally symmetric, which makes fully spherical convolutions redundant. We therefore improve the time and space efficiency of the RT-ESD convolution by reducing the spherical graph G to a hemispherical graph H . Because f is antipodal symmetric, we assume, without loss of generality, that V is a symmetric sampling, i.e. ∀ p ∈ V , ∃ ( -p ) ∈ V . We construct the hemispherical sampling V + = { p ∈ V and p z ≥ 0 } from the spherical sampling V , with p z chosen to be the 3rd spatial coordinate of p . We compute the hemispherical Laplacian weight as,

Thus, the spherical operation L f is entirely explained by the hemisphere graph Laplacian L + f + , i.e. ∀ p ∈ V , ∃ ( p + ) ∈ V + s.t. ( L + f )( p + ) = ( L f )( p ) , where f + is the f sampling on V + (proof

Reconstruction

Loss

Normalization

Equivariant Hemi-Spherical U-Net

Reconstruction

Spatio-Hemispherical dMRI

Vertices p,q

[A] Hemispherical Graph Construction

Spherical Graph

Hemispherical Graph

Vertices

Edges

p

q

Edges

Vertices

p

q

-p

Raw dMRI

[B] fODF Estimation Framework

Predicted fODF

Sparsity loss

Non-Neg. loss

Spatial loss

Convolution

+ Batch-Norm

+ ReLU

k: Feature multiplicator

Mean-Pooling

Upsampling

Concatenation

Hemisphere Reduction

L(p,q)

L+(p,q)

LERxV

L+ ∈RVxV

=

=

b

q

in appendix A.1). For spherical samplings V such as HEALPix, our proposed sampling reduces the number of vertices sampled to 1 2 |V| and the hemispherical Laplacian size to 1 4 |V| 2 . Our reduction is both theoretically and empirically equivariant to E ( 3 ) × SO ( 3 ) , see appendix A.2 for more details.

Dense Matrix Multiplication. DeepSphere's [53] used sparse matrix multiplication to compute T k ( L ) f ( x ) . However, for dMRI, our Laplacians are dense and sparse matrix multiplication adds significant computational overhead on dense matrices. To address this, we substitute the sparse matrix multiplication used in [53] with standard matrix multiplications and find substantial efficiency improvements when applied to dMRI data.

Pre-computed Chebyshev Polynomials. DeepSphere [53] assumes the spherical sampling V , and thus the Laplacian L , to be dependent on the input spherical signal f . Thus, for efficiency, the Chebyshev polynomials are computed iteratively as T k +1 ( L ) f ( x ) = (2 L T k ( L ) -T k +1 ( L )) f ( x ) for each new f and L . In our setting, we use the same spherical sampling V for every spherical signal f , making the Laplacian L of every convolutional layer fixed. We therefore precompute and store these polynomials before training to further eliminate redundancy.

3.3 Spatial Hemispherical Deconvolution (SHD) Network

Fig.3 presents an overview of our proposed spatial-hemispherical deconvolution (SHD) framework. Below, we first describe how our inputs are preprocessed and then detail our network and the losses used to train it.

Data normalization. The dMRI signal acquisition yields a raw spatio-spherical signal S G sampled on a set of spherical coordinates G , called shell sampling. The signal intensity range and shell sampling are scan-dependent and need to be normalized before being fed to our proposed neural network. Following [55], we normalize the dMRI scans such that the white matter B 0 -diffusion signal is comparable between scans. Then, following [19], the normalized data is interpolated onto a hemispherical HEALPix sampling V + . We provide details in appendix C.2.

E ( 3 ) × SO ( 3 ) -Unet. The interpolated diffusion signal is then deconvolved to recover the spatio-spherical fODFs F V + on the same input HEALPix sampling V + . We use a U-Net, with 4 depth levels, with layers acting on a spatial grid of hemispherical samples. We use our proposed layers in the U-Net architecture presented in Fig. 3. Besides the last, every block comprises a convolution, batch-norm, and a ReLU activation. The last block consists of a convolution and a Softplus activation function. The up/downsampling layers first involve mean spatial upsampling/pooling on R 3 , followed by mean spherical upsampling/pooling on S 2 , both introducing minor numerical equivariance error. Following DeepSphere [53], to minimize the equivariance error from the spherical upsampling/pooling, we use the hierarchical structure of the HEALPix grid.

Network training. Our proposed deconvolution network N θ takes as input the shell-sampled dMRI signal S G to produce fODFs F θ V + = N θ ( S G ) . We train the network in an unsupervised manner by reconstructing the dMRI signal S θ D = C ( F θ V + ) from the estimated fODFs, with details on the

RT-ESD

[A] Runtime

as percentage of Baseline method

PONITA

HemiSpatio

100

50

150

200

U-Net

Inference runtime (%)

200

100

12

48

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768

U-Net

300

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[B] Memory usage

as percentage of Baseline method

Inference memory usage (%)

Single layer

Spherical Graph Resolution (number of spherical vertices)

Single layer

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12

48

192

768

Spherical Graph Resolution (number of spherical vertices)

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U-Net

U-Net

Single layer

Single layer

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768

Spherical Graph Resolution (number of spherical vertices)

40

20

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192

768

Spherical Graph Resolution (number of spherical vertices)

Ablation

Baseline

+ Hemisphere

+ Dense

+ Precomputed Poly.

0

Inference runtime (%)

Baseline

+ Hemisphere

+ Dense

+ Precomputed Poly.

80

60

100

40

20

0

80

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40

20

0

Inference memory usage (%)

Ablation

Baseline

+ Hemisphere

+ Dense

+ Precomputed Poly.

[C] Runtime ablation

as percentage of Baseline method

[D] Memory ablation

usage as percentage of Baseline method

RT-ESD

PONITA

SHD (Ours)

Convolution

RT-ESD

PONITA

SHD (Ours)

Convolution

RT-ESD

PONITA

HemiSpatio

Baseline

+ Hemisphere

+ Dense

+ Precomputed Poly.

X

X

+

reconstruction function C in appendix C.2. We note that the input and reconstruction shell samplings G and D need not be the same, allowing for angular super-resolution reconstruction during training if required. The parameters θ are estimated by minimizing a reconstruction loss subject to non-negativity [19, 66] and sparsity-promoting regularization [12, 19]:

where the terms correspond to reconstruction, non-negativity, sparsity, and smoothness, respectively. We define A as a vector setting any positive component of F θ V + to 0. We set σ = 10 -5 following [12]. L TV is described below.

Spatial regularization. To add explicit spatial regularization, we propose to extend the spatial totalvariation regularization from [11] to the trainable model framework L tv = ||∇ x F θ V + || 2 2 where ∇ x is the spatial gradient operator. The spatial regularizer L tv seeks to promote smoother reconstructions in ill-posed settings, especially at clinical angular resolutions used later in our experiments.

4 Experiments

We first quantify the runtime and memory efficiency gains produced by our contributions. We then analyze their use across a diversity of dMRI deconvolution settings on both real and synthetic datasets.

4.1 Runtime and memory efficiency improvements

Experimental details. Wefirst compare the efficiency of our proposed convolution against two spatiospherical equivariant convolutions: RT-ESD [20] and PONITA [9], as measured by inference runtime and memory consumption. We then conduct an ablation analysis of our three proposed improvements. All analyses are conducted for both a single spatio-spherical layer applied to increasing spherical graph resolution and also for an entire spatio-spherical U-Net architecture detailed in Fig. 3. As input, we use a random 8 × 8 × 8 × V spatio-spherical volume, with V depending on the spherical HEALPix grid resolution. We fix the U-Net input HEALPix resolution to 8 ( V = 754 ) and we vary the single layer HEALPix resolution from 1 ( V = 12 ) to 8 . We take the RT-ESD convolutions as a reference baseline and present runtime and memory usage results as a percentage of the baseline averaged across 50 inference steps. All comparisons are performed using an Nvidia RTX-8000 GPU.

[A] Dataset I and II : fODF estimation

Generalization to unseen volume

Super-Angular fODF estimation

dMRI dataset

DiSCo

HCP

fODF

Reconstruction

Model input

Exp. 1:

Exp. 2:

high-resolution

Exp. 1 & 2:

low-resolution

high-resolution

Unsupervised fODF estimation

dMRI dataset

Tractometer

Tractography

fODF

Exp. 2:

Exp. 1:

Model input

Reconstruction

low-resolution

low-resolution

Results. Fig. 4 presents the results of the efficiency analysis. For a single layer, our convolutions are 2 to 5 times faster than the RT-ESD baseline while decreasing memory footprint by 2 . 5 times on large spherical sampling. PONITA has a similar runtime and memory footprint as SHD on sparse spherical sampling but underperforms on large spherical graphs (that are common in dMRI deconvolution [12]), being 10 times slower and more memory intensive. Consequently, a U-Net designed for highangular resolution sampling is 2 . 5 and 3 . 5 faster using our proposed convolutions over RT-ESD and PONITA, while using 4 and 20 times less memory, respectively. Our ablations indicate that using our hemispherical convolutions over the spherical convolutions, as well as using precomputed Laplacian polynomials, brings the most improvement for both runtime and memory on large spherical sampling while leveraging dense matrix multiplication reduces runtime on low-angular resolutions.

4.2 Diffusion MRI deconvolution experiments

Experimental details. Validation of fODF estimation methods is confounded by the absence of known and unbiased ground truth fODFs in in vivo human dMRI. To address this limitation, the literature commonly benchmarks methods on synthetic datasets with known ground truth tissue microstructure. We conduct both a quantitative (synthetic dataset) and qualitative ( in vivo human dMRI) analysis of our deconvolution framework (Section 4.2.1). We then perform a quantitative analysis on the tractography downstream task on a benchmark synthetic dataset (Section 4.2.2). For quantitative evaluation, we first estimate fODFs, then detect fiber peaks, and report the peak angular error and false positive rate (FPR) following [17]. The angular error is the average angle between the ground truth peaks and their closest predicted peaks and the FPR is the number of predicted peaks that do not match any ground-truth peaks divided by the number of ground-truth peaks. See Appendix A.3 for more details. All deep network experiments were performed using a single RTX8000 GPU and used less than 16 GB of system RAM.

Baselines. We compare the proposed SHD method against conventional iterative methods including CSD [66], RUMBA and RUMBA-TV [11], and state-of-the-art learning based methods leveraging different convolutional layer, such as the non-equivariant voxel-wise MLP in [52] and the volume-wise CNN in [46], that we also extend to have spatial regularization ( CNN-TV ). We also compare against equivariant voxel-wise spherical convolutions ESD [61, 19], and more recent equivariant spatiospherical convolutions RT-ESD [20] and PONITA [9]. To focus on the inductive bias introduced by each method's convolution layer, we adopt a standardized network architecture and training framework, presented in section 3.3. We provide details about these different baselines in appendix C.3.

Data. We leverage three public dMRI datasets for validation. DiSCo [54] is a synthetic dMRI dataset with three volumes, split into training, validation, and testing volumes, with high-angular resolution sampling ( 4 shells each with 90 gradients) provided with ground truth fODF. We then use 100 in-vivo unrelated dMRI scans from the HCP young adult dataset [70], split into 65 training, 15 validation, and 20 testing volumes, with high-angular resolution ( 3 shells, each with 90 gradients), without any ground truth. As we aim to benchmark performance on both high-angular resolution data and more commonly acquired low-angular resolution data, we further simulate low-angular

[A] DiSCo fODF:

High-angular resolution input

[B]

DiSCo fODF:

Low-angular resolution input

Angular Error ° (  )

Angular Error ° (  )

[C]

2-Crossing fiber:

High-angular resolution input

[D]

2-Crossing fiber:

Low-angular resolution input

SHD-TV

(ours)

SHD

(ours)

RT-ESD

PONITA

ESD

CNN-TV

CNN

MLP

CSD

RUMBA-TV

RUMBA

18

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26

0.20

0.15

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False Positive Rate (  )

Conventional

Not Equivariant

SO(3)

-Equivariant

SE(3)

-Equivariant

E(3)xSO(3)

-Equivariant

RT-ESD

PONITA

ESD

RUMBA

RUMBA-TV

CSD

MLP

CNN

CNN-TV

SHD-TV

(ours)

SHD

(ours)

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0.40

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False Positive Rate (  )

20

26

28

30

32

22

24

34

Conventional

Not Equivariant

SO(3)

-Equivariant

SE(3)

-Equivariant

E(3)xSO(3)

-Equivariant

Input

Sampling

CSD

ESD

CNN

SHD-TV (ours)

Ground-Truth

Input

Sampling

CSD

ESD

CNN

SHD-TV (ours)

Ground-Truth

resolution acquisitions on DiSCo and HCP by randomly selecting 29 gradients from only the lowest shell. The last dataset is Tractometer [49], a single-volume simulation of a real human brain with a low-angular resolution protocol ( 1 shell, 32 gradients), provided with ground truth tractography alongside a tractography scoring algorithm [60].

4.2.1 Super-resolved fODF estimation: synthetic DiSCo and real human HCP datasets

We perform two experiments, depicted in Fig.5A). First, we measure performance when all baselines are trained and tested on high-angular resolution data. Second, we give all methods high-angular resolution data only during training and test them on low-angular resolution data. In this setting, the networks are trained to regress the high-angular resolution data from low-angular resolution input. For quantitative evaluation, we extract fiber directions from the estimated fODF and we report the peak angular error and false positive rate as described in the previous section. All results are averaged over five random training seeds. Finally, as HCP lacks fODF and tractogram ground truth, we perform a qualitative analysis. Further validation details are in appendix A.3 and A.4.

Results. Fig. 6 presents DiSCo deconvolution results for both the low and high-angular resolution settings. We visualize qualitative results on a random HCP test subject in Fig.2, focusing on two areas with diverse tissue microstructures, such as crossing fibers, fiber bending, and multi-tissue compartments. Our proposed efficiency contributions retain all of the performance of previous methods as our proposed SHD model (without spatial regularization) achieves similar results to the much slower and more memory-intensive RT-ESD and PONITA methods. Moreover, non-equivariant methods, while having competitive angular errors, are negatively impacted by a high false positive rate and qualitatively high number of spurious fibers.

Finally, on research-grade high-angular resolution data, incorporating spatial information in the network input does not improve results. However, adding total-variation regularization on the output of the model as in our proposed SHD-TV model shows significantly improved results. Further, in a clinical low-angular setting, our spatially informed models ( SHD and SHD-TV ) widely outperform voxel-wise methods, showcasing the importance of contextual information for clinical dMRI.

4.2.2 Unsupervised fODF estimation and tractography w/ known ground truth: Tractometer

A high-level overview of this experiment is described in Fig.5B). As Tractometer consists of a single volume with a low-angular resolution protocol with no held-out testing data, we benchmark methods on unsupervised fODF estimation similarly to previous work [19]. We first train the models in an

[C] Tractometer:

Ground-Truth

RUMBA

ESD

SHD (ours)

SHD-TV (ours)

[A] Tractometer:

Unsupervised fODF estimation

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(ours)

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(ours)

RT-ESD

ESD

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RUMBA

18

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Angular Error (  )

16

SHD-TV

(ours)

SHD

(ours)

RT-ESD

MLP

ESD

CSD

RUMBA

RUMBA-TV

CNN

CNN-TV

73

75

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79

74

76

78

72

0.60

0.55

0.50

0.45

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[B] Tractometer:

PONITA

PONITA

26

Conventional

Not Equivariant

SO(3)

-Equivariant

SE(3)

-Equivariant

E(3)xSO(3)

-Equivariant

Conventional

Not Equivariant

SO(3)

-Equivariant

SE(3)

-Equivariant

E(3)xSO(3)

-Equivariant

unsupervised manner to estimate fODFs and then use them to investigate the effect of improved local fODF estimation on white matter tractography accuracy. Validation of the unsupervised estimated fODFs involves using the same local fiber detection evaluation as for the DiSCo experiments. Subsequently, the validation of the estimated tractographs is conducted by comparing them against 21 ground truth fiber bundles provided by the dataset [60]. The white matter streamlines of the tractograph are initially segmented into 21 target fiber bundles. The correctly identified bundles are counted as Valid Bundles and the streamlines belonging to one of the ground truth bundles are counted as Valid Streamlines. Further, the volume of each estimated bundle is compared against the volume of its corresponding ground truth bundle, leading to the computation of the Overlap score (analogous to a True Positive value) and the Overreach Score (similar to a False Positive value).

Results. We present Tractometer results in Fig.7. In this setting where all methods are trained and tested on the same volume, non-equivariant methods ( CNN and CNN-TV ) outperform most unregularized equivariant deconvolution methods. We speculate that when generalization is not required, non-equivariant models enjoy a higher overfitting capability in this problem, thereby diminishing the advantage of employing inductive priors in neural networks.

However, the proposed SHD-TV , which incorporates both spatio-spherical inductive biases and spatial regularization, significantly outperforms all other methods, regardless of whether they are equivariant or spatially regularized. This finding supports the compounding benefits of both our architectural modifications and our proposed regularization strategy. In turn, these enhanced fODF recovery performances positively impact tractography accuracy, reflected in a quantitatively higher number of valid streamlines and bundle overlap (albeit with an increase in bundle overreach) and more qualitatively coherent streamlines.

5 Discussion

Limitations and future work. Our work has certain limitations. First, while we focus on the key dMRI task of fiber deconvolution, our proposed layers are task-agnostic and can also be deployed to other tasks such as denoising and segmentation in future studies. Second, while our layers can be used in many fields with spatio-spherical data outside of neuroimaging, such as robotics [72], neural rendering [26], and molecular dynamics [9], our assumption of antipodal symmetry (widespread in dMRI) may need to be relaxed for many applications within these fields. Third, angularly

undersampled reconstruction results have an inherent risk of missing fibers or fiber hallucination. While our work partially mitigates these unrealistic reconstructions by adding geometric and spatial priors to the reconstruction, future validation studies could analyze cohorts with paired scans on clinical low-resolution and research-grade high-resolution dMRI protocols. Finally, our methods need further testing on additional clinical challenges such as subject motion, imaging artifacts, and abnormal structures. However, as our work is entirely self-supervised and only based on regularized reconstruction of input data, we anticipate robustness to such imaging variation. Preliminary results on a pathological brain sample are provided in appendix A.5.

Conclusions. The analysis of in vivo white matter neuronal organization within the brain depends on the key task of dMRI deconvolution and current methods are either not accurate or too slow. To that end, we developed highly time and memory-efficient equivariant convolutional network layers that respect the physical symmetries of dMRI. Our contributions led to 3 . 5 × faster runtimes and up to 20 × lower memory consumption as compared to existing spatio-spherical layers, while exceeding or maintaining their performance on dMRI studies. These layers were then used to construct networks that achieved strong dMRI deconvolution performance in practical timeframes across multiple commonly used synthetic benchmark datasets and real in vivo human datasets. Finally, our methods are robust to the dMRI angular resolutions typically used in clinical practice outside of research settings, potentially enabling robust analyses of large-scale retrospective dMRI studies and direct application to the clinic.

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A Additional results

A.1 Spherical and hemisphere convolution

Hemispherical sampling. Consider the symmetric spherical sampling V = { p ∈ S 2 } such that ∀ p ∈ V , ∃ ( -p ) ∈ V , and the Euclidean coordinate p = ( p x , p y , p z ) ∈ S 2 . The sphere can be divided into a north and south hemisphere V = V + ∪ V -, with V + = { p ∈ V , such that p z > 0) } and V -= { p ∈ V , such that p z < 0) } . For point p lying on the circle p z = 0 , we further add to V + every point such that p y > 0 and to V -every point such that p y < 0 . Finally, for the two points p such that p z = p y = 0 , we add p such that p x > 0 to V + and p such that p x < 0 to V -.

Hemispherical restriction. Consider a spherical function f : S 2 → R and a function L : S 2 ×S 2 → R , both sampled on V such that f ∈ R V and L ∈ R V × V . Consider the function L + : S 2 ×S 2 → R , L + ( p, q ) = L ( p, q ) + L ( p, -q ) , sampled on V + such that L + ∈ R V/ 2 × V/ 2 and f + ∈ R V/ 2 the sampling of f on V + .

Theorem 1 Consider an antipodally symmetric spherical function f and L , such that for all p, q ∈ V , f ( p ) = f ( -p ) and L ( p, q ) = L ( -p, -q ) , then ∀ p ∈ V , ∃ p + ∈ V + , such that ( L f )( p ) = ( L + f + )( p + ) .

Proof Let p ∈ V . Then, define p + = p if p ∈ V + , -p otherwise. Thus, we know that p + ∈ V + .

(12)

A.2 Numerical equivariance error analysis

Motivation. Due to discretization and implementation practicalities, equivariant networks are subject to aliasing and are only approximately equivariant [33]. Prior work in this space [53, 19, 20] has demonstrated low equivariance errors for SO ( 3 ) and E ( 3 ) × SO ( 3 ) convolutions and their U-Net counterparts when operating on full spherical sampling. We claim that the proposed accelerated implementations of these convolutions do not introduce additional equivariance error and provide empirical evidence below.

Model details. The hemispherical E ( 3 ) × SO ( 3 ) convolution is compared to a grid-wise E ( 3 ) -equivariant convolution with spherical information, E ( 3 ) -SH, and a voxel-wise SO ( 3 ) -equivariant convolution with spatial information, ConcatSO ( 3 ) . The E ( 3 ) -SH convolution is a 3D convolution with isotropic kernels processing spherical harmonic coefficients as input features [52]. The ConcatSO ( 3 ) convolution is a spherical convolution processing spatial neighbors as input features [62, 20]. The spatial and spherical kernels are initialized randomly following [34].

Evaluation. We measure numerical equivariance error as the deviation from the equivariance equality, using:

where N is the tested operator, f is a spatio-spherical signal, and G = ( T, R ) ∈ E ( 3 ) × SO ( 3 ) is a composition of grid-wise and voxel-wise transformation. In practice we only test rotationequivariance, and we limit T and R to be rotations. This quantity is measured over 1000 randomly sampled spatio-spherical signals f , random rotations ( T, R ) , and randomly initialized N . f is randomly generated from a uniform U [0 , 1] distribution on a 8 × 8 × 8 × V spatio-spherical volume with HEALPix hemispherical grid of resolution 8 ( V = 384 vectors). Moreover, we filter high spherical frequency by only keeping the first 8 spherical harmonic degrees.

Results. Fig.8 illustrates estimated equivariance errors. As expected, we find that the E ( 3 ) -SH convolution exhibits a lack of equivariance to voxel-wise rotations, and the ConcatSO ( 3 ) convolution exhibits a lack of equivariance to grid-wise rotations. The accelerated implementation of the E ( 3 ) × SO ( 3 ) convolution, taking into account both inductive bias of the spatial and spherical domain, maintains low voxel-wise rotation equivariance errors.

Equivariance Error in Function of Convolution Equivariance Group

Equivariance Error

0.00

2.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

< 0.01

Layer

Type

E(3)x SO(3)

Convolution

SO(3)

Convolution

< 0.01

U-Net

E(3)

-SH

Convolution

Grid

Rotation

Voxel

Rotation

Both

Rotation

Grid

Rotation

Voxel

Rotation

Both

Rotation

Grid

Rotation

Voxel

Rotation

Both

Rotation

A.3 DiSCo quantitative results

Experiment details. We provide more details on the DiSCo dataset, which we used to evaluate both fODF estimation performance and robustness to dMRI protocol angular resolution. The DiSCo dataset comprises three 40 × 40 × 40 synthetic volumes with variable amounts of synthetic noise added. Our experiments use the volumes with a noise level of SNR= 30 . All three volumes share the same synthetic generation process and have four B0 images and four shells each with 90 gradients. We do not apply any pre-processing step. For quantitative analysis, we train the different models on the first volume, validate on the second, and test on the third. Given that the synthetic generation process of the DiSCo dataset is similar to a white matter/CSF tissue simulation, we limit the different models to a 2 tissue decomposition, and we analyze the white-matter fODFs.

Validation details. All results for deep network models are averaged over five random seeds. For quantitative evaluation, we first estimate fODFs, then detect peaks, and finally threshold them based on their voxel-wise relative fiber partial volume applying 19 different threshold values ranging from 0 . 05 to 0 . 95 . For each threshold, we calculate the false positive rate (FP), false negative rate (FN), and angular precision of detected fibers following the evaluation framework proposed by [17]; utilizing a rejection cone of 25 ◦ . Precision and recall are computed at different thresholds using FP, FN, and true positive fibers (TP). Subsequently, we calculate the Precision-Recall F1 score. We report the angular error and false positive rate for the threshold that maximizes the F1 score on the DiSCo validation volume. We report raw average scores and 95% confidence interval (CI) average over five random seeds in Table 1. Notice that all scores have a CI < 0 . 01 , but the angular error.

A.4 Qualitative in vivo human results

Experiment details. We provide more details on the HCP dataset, which we used for qualitative evaluation of the fODF estimation. The HCP young adult release contains 100 unrelated subjects and

ensures the exclusion of twin subjects to prevent any data leakage between dataset splits. The dMRI volumes had undergone pre-processing using FSL [36] and FreeSurfer [25], and further details on the preprocessing pipeline can be found in the HCP dataset documentation [28, 2, 3, 4]. The 100 dMRI volumes adhered to a high-resolution diffusion protocol, featuring 18 B0 volumes per subject and three diffusion shells, each comprising 90 diffusion gradients. We apply similar downsampling to the HCP dataset, randomly selecting 29 diffusion gradients from the lower b-value shell ( 1000 , s/mm 2 ) while retaining the B0 volumes. For qualitative analysis on the HCP dataset, we train on 65 randomly selected subjects, monitor training on 15 validation subjects, and test on 20 subjects. In contrast with the DiSCo dataset, we estimated 3 group-wise tissue response functions, accounting for white matter, gray matter, and cerebrospinal fluid present in in-vivo human brains.

Validation details. In-vivo datasets lack microstructure ground truth. Some approaches [62] suggest using the CSD estimated fODFs from high-quality and high-resolution dMRI as a surrogate ground truth. However, we argue that this introduces a bias towards CSD and hinders the assessment of performance improvements in new methodologies. A more recent alternative proposed by [19, 20] involves using partial volume estimation (PVE) from fODFs as a validation metric. Nevertheless, the PVE scalar is rotation invariant, posing limitations on its ability to discern the impact of rotationequivariant methods compared to their non-equivariant counterparts. Non-equivariant models may demonstrate high accuracy in invariant scalar estimation while failing to capture the underlying geometric structure adequately. Moreover, PVE validation relies heavily on the accuracy of thirdparty tissue segmentation on T1/T2 images, which may not consistently correlate with partial volume estimates derived from more information-rich but low-resolution dMRI data. While we acknowledge that a comprehensive solution for quantitative analysis of in-vivo fODF estimation is yet to be established, we advocate for a visual qualitative analysis of our proposed method. This approach aims to provide insights into the performance of our model without relying on potentially biased or limited validation metrics.

A.1) CSD

Multi-crossing area

A.1) CSD

3-Shell High-Angular Resolution

1-Shell Low-Angular Resolution

Cerebellum

B.1) ESD

Multi-crossing area

B.1) ESD

3-Shell High-Angular Resolution

1-Shell Low-Angular Resolution

Cerebellum

C.1) CNN

Multi-crossing area

3-Shell High-Angular Resolution

1-Shell Low-Angular Resolution

Cerebellum

D.1) SHD (ours)

Multi-crossing area

D.1) SHD (ours)

3-Shell High-Angular Resolution

1-Shell Low-Angular Resolution

E.1) SHD-TV (ours)

E.1) SHD-TV (ours)

Multi-crossing area

3-Shell High-Angular Resolution

1-Shell Low-Angular Resolution

Cerebellum

A.5 Preliminary analysis on an abnormal brain

Here, we demonstrate the preliminary applicability of our method on abnormal brains. In Fig. 10, we estimate fODFs on a brain with a glioma using the recently released dMRI dataset from [75]. We find that the additional priors from our method help fODF estimation and fiber tracking substantially, filling in a hole in the fiber tracks recovered using the baseline CSD method. However, we note that this analysis is highly preliminary and that tractography on brains with lesions is a highly active area of research that requires substantial modifications to tractography algorithms [77].

[A] Subject with glioma

[A.1] CSD

[A.2] SHD-TV (ours)

Crossing-fiber area and

Crossing-fiber area and

tumor border

tumor border

Tractography

Tractography

B Equivariance vs. Data Augmentation: R 3 ×S 2 MNIST

Motivation. We study the advantages of using spatio-spherical inductive biases for convolutional layers, as opposed to learning them from the data. We create a synthetic spatio-spherical MNIST dataset (inspired by the spherical MNIST experiments of [15]) to conduct controlled experiments.

Dataset. The R 3 ×S 2 MNIST data generation process is presented in Fig.11. Importantly, the dataset characterizes a spatio-spherical segmentation task, where voxel-wise information alone is insufficient for correct voxel classification. We construct two versions of this dataset to test generalization from either equivariance or data augmentation. The first version lacks any form of data augmentation, whereas the second version incorporates random grid and voxel-wise rotations. Because on-the-fly data augmentation of spatio-spherical signals is computationally expensive, we pre-generate a total of 1000 volumes per dataset, with a training-validation-test split of 716 / 142 / 142 .

Dataset generation. We randomly position eight non-overlapping 4 × 4 squares on a 2D 16 × 16 slice. Subsequently, each square is assigned a random digit between 1 and 9, designating 0 as the background digit. This 2D slice is duplicated along the z -axis, yielding the final 3D segmentation volume. Notably, this volume comprises eight non-overlapping 4 × 4 × 16 structures, termed tubes , aligned along the z -axis, each featuring the same classification digit across its voxels. Optionally, we apply a random grid rotation to the segmentation volume. In the second step, we randomly select an MNIST image corresponding to the classification digit assigned to each square. These digit images are projected onto a sphere using the methodology presented in [15]. We utilize HEALPix spherical sampling with a resolution of 4 , equivalent to a spherical resolution of V = 192 vertices. Given the straightforward nature of voxelwise spherical digit classification, we reduce the amount of information at each voxel. Instead of projecting the entire MNIST image onto the sphere, we randomly crop it to one-quarter of its original size before projecting the cropped digit onto the sphere. As a result, any segmentation network has to rely on both the spatial neighborhood and the spherical information within a voxel. Optionally, we apply a random voxel rotation to the spherically projected digit.

A)SyntheticVolume Generation

B)Voxel-wise MNIST Spherical Projection

C) R²× S²MNIST

Label generation

MNISTRandom crop

Spherical Projection,

Spherical MNIST

R3 Grid

s2Voxel

Rotation

Rotation

D) Segmentation Dice Score (↑)

E) Segmentation Dice Score (↑)

No rotation during training

Random R? × S2 rotation during training

个

Dice Score (% -

1

During testing:

1

0.75

Norotation

0.75

R3×S²rotation

Dice Score

0.5

0.5

During testing:

0.25

0.25

Norotation

R3×S²rotation

0

0

SO(3)

Concat-SO(3)

E(3)-SH

SO(3)

Concat-SO(3)

E(3)-SH

E(3)×SO(3)

Convolution

Convolution

Convolution

Convolution

Convolution

Convolution

Convolution

Convolution

Model details. We compare four convolutional architectures, each exhibiting different equivariance properties while sharing the same U-Net structure presented in Fig.3. We compare the three previously presented spatio-spherical E ( 3 ) × SO ( 3 ) , E ( 3 ) -SH, and ConcatSO ( 3 ) U-Nets, and, to highlight the importance of incorporating spatial information, we compare them with a voxel-wise SO ( 3 ) U-Net. Inputs to the U-Nets consist of 16 × 16 × 16 × 192 R 3 ×S 2 MNIST volumes, and the outputs,

post-Softmax activation, are 16 × 16 × 16 × 10 volumes representing the classification probabilities of each voxel for the 10 digits.

Training details. Our training regimen uses a batch size of 8 and trains for 50 epochs with an initial learning rate of 3 × 10 -3 , which is halved after the 25 , 35 , and 45 epochs. The optimized objective function encompasses a combined Dice and Cross-Entropy loss with class-dependent weight to address imbalanced labels.

Results. We first assess how equivariance influences the generalization of models to unseen data transformations. The four models are first trained on the dataset without data augmentation and subsequently tested on the dataset incorporating out-of-distribution random rotations. We then train and test the models on the dataset augmented with random rotations. The results are presented in Fig.11.D) and E). To quantify segmentation performance, we compute the dice score across every volume in the testing set.

In Fig.11.D), we observe that all four models, regardless of the embedded equivariance, perform well when trained and tested on a transformation-free (i.e., in-distribution) dataset. Importantly, the spatially informed networks outperform the voxel-wise SO ( 3 ) network, underscoring the significance of incorporating spatial context into the neural network. Furthermore, we emphasize the importance of incorporating inductive bias into the neural network to enhance generalization to unseen data transformations during training. The E ( 3 ) × SO ( 3 ) model, trained on the rotation-free dataset, achieves a relatively better dice score of 0 . 38 on the dataset with unseen rotations than both other models with only equivariance to E ( 3 ) or SO ( 3 ) .

Fig.11.E) further illustrates the advantage of inductive bias compared to data augmentation during training. Data augmentation dramatically enhances the performance of the E ( 3 ) × SO ( 3 ) model on randomly rotated data, surpassing its non-equivariant counterparts. Meanwhile, data augmentation applied to E ( 3 ) or SO ( 3 ) models lacking spherical or spatial inductive bias improves segmentation performance but falls short of matching the performance of the E ( 3 ) × SO ( 3 ) equivariant model.

C Additional experimental details

C.1 Spatio-Hemispherical Convolution overview

A)Input Neighborhood

B)Voxel-wise Hemispherical Filtering

C)Filtered Neighborhood

→

T°(L)f(x)

ERFin

Graph Filtering

Concatenation

E

RFinK

...

>

TK-1(L)f(x)

f:R3→RVxFin

f(x)∈RVxFin

目

ERFin

f(x)∈RVxFinK

f:R3→RVxFinK

V:hemispherical vertice

E)Vertex-wise Spatial Convolution

F) Convolution Output

c,y:spatialvoxel coordinates

IsotropicWeights

f,f,foutut,ltreutut i

α:R→RFinKxFout

f(z),f(z): spherical signal at voxel x

→

。

fu,fout, : spatial signal at vertex v

Convolution

L:laplacian of hemispherical graph

∑α(lle -yll)f(x)

Ti：th chebyshev polynomial

f:R3→RVxFinK

2EP

fout : R3→RVxFout

K:polynomial degree

f : R3→RFinK

fout,y : R3 → RFout

v:hemispherical vertex

Fin,Fout:input and output features

C.2 Details of the deconvolution framework

fODF model. The fODF model assumes the voxel-wise diffusion signal to be produced by a composition of T tissue types, with each tissue further composed of an unknown number of cells/fibers, each one described by its local partial volume and orientation. At the voxel level, the fiber information is aggregated into one fiber orientation distribution function (fODF) per tissue, providing information on the local tissue composition and orientations. The model relates the dMRI signal and tissue microstructure by combining T tissue-specific Response Functions (RFs) and T fiber Orientation Distribution Functions (fODFs). The fODFs F = { F t } are tissue-dependent antipodally symmetric spatio-spherical functions. The response functions RF = { RF b t } are tissue-dependent and shelldependent spherical functions, assumed constant across the spatial domain, and to be antipodal and axial symmetric. The total b -shell dMRI signal is modeled as a sum over tissue-specific signals S b = ∑ T t =1 S b t , where tissue-specific diffusion signals are modeled as a voxel-wise spherical convolution S b t = C ( F t , RF b t ) between the fODF and the response function. The convolution is defined as C ( F, RF )( x, q ) = ∫ p ∈S 2 F ( x, p ) RF ( q T p ) d p .

Diffusion sampling. A dMRI image S is a function S ( x, q ) : R 3 ×S 2 → R B . In practice, the dMRI signal is discretized on a set of shells B = { b i ∈ R + } i ∈ [1 ,.,B ] where B is the number of shells, such that for every b ∈ B , the b -shell function S b is a spatio-spherical signal. Furthermore, the b -shell function is sampled on a shell-dependent set of spherical gradient direction V b = { q b i ∈ S 2 } i ∈ [1 ,.,N b ] with N b the number of b -shell gradient sampling. We note S b , V b ( x ) = [ S ( x, q b 1 , b ) , ..., S ( x, q b N b , b )] ∈ R N b the sampled b -shell signal on V b . Moreover, we note D = {B , {V b } b ∈B } the full diffusion sampling set, and S D ( x ) = [ S b 1 , V b 1 ( x ) , ..., S b B , V b B ( x )] ∈ R N D the sampled dMRI signal, with N D = ∑ b ∈B N b the total number of gradient sampling.

Data normalization. For every new dMRI scan S ori. , we compute the scan-specific white matter B0 response function RF 0 ,ori. wm using MRtrix [68], and normalize the dMRI signal S = S ori. /RF 0 ,ori. wm . The input of our framework is the normalized shell-sampled diffusion MRI signal S G ( x ) = [ S b 1 , V b 1 ( x ) , ..., S b B , V b B ( x )] ∈ R N D for x ∈ R 3 . Following [19], we first normalize the protocoldependent spherical sampling by first interpolating it, for each diffusion shell, to a fixed hemisphere HEALPix sampling V + as defined in section 3.2. We implement the interpolation using spherical harmonics. For this, we first compute the spherical harmonic coefficients ˆ S b ( x ) ∈ R N L for every

b ∈ B shell, where N L is the number of harmonic coefficient for a maximal harmonic bandwidth of L . Because the diffusion signal is antipodal-symmetric, the odd-order spherical harmonic coefficients are equal to 0 and we only compute the even-order coefficients, making the total number of coefficient N L = ( L/ 2 + 1)( L +1) . Importantly, getting the coefficients ˆ S b ( x ) from S b , V b ( x ) requires at least |V b | ≥ ( L/ 2 + 1)( L +1) samples. We limit the maximum interpolation bandwidth to L = 8 , which requires at least 45 samples per shell. In case the shell sampling V b does not have enough samples, we use the theoretically maximum spherical harmonic degree estimated from the available gradients. We then interpolate the harmonic coefficients on the HEALPix grid V + , S b , V + ( x ) = ˆ S b ( x ) Y L V + , where Y L V + relates the even L bandwidth harmonic coefficients to the HEALPix spherical sampling. We get the B feature input spatio-spherical signal S V + ( x ) ∈ R V × B .

Group-average response function. We estimate a population-based response function ˆ RF following [55]. For N s normalized dMRI scans in our training dataset, we first compute T tissue response functions ˆ RF t,i per dMRI image, where i is the image index, using the Dhollander algorithm, implemented in MRtrix [68] [18]. We then compute the per-tissue average response function ˆ RF t = 1 /N s ∑ N s i =1 ˆ RF t,i .

Signal reconstruction. From the fODFs F V + , we reconstruct the diffusion signal S D on the reconstruction shell-sampling D using a spherical convolution with response functions RF . For every voxel x ∈ R 3 , we compute the spherical harmonic coefficients ˆ F ( x ) = { f m l,t ( x ) } ∈ R N L × T from the estimated F V + ( x ) . The fODFs being antipodal-symmetric, we only compute the even-degree coefficients. The maximum spherical harmonic degree L depends on the resolution of the HEALPix sampling V + . We use a sampling resolution with 384 vertices per hemisphere and a fODF maximum spherical harmonic degree of L = 18 . Moreover, the response functions are also represented by their spherical harmonic coefficients ˆ RF = { r m,b l,t } ∈ R M L × T × B . The RFs are antipodal and z -axis symmetric, setting to 0 every non-even degree and non-zero order coefficients. We then compute the reconstruction dMRI spherical harmonic coefficients ˆ S ( x ) = { s m,b l ( x ) } ∈ R N L × B , s m,b l ( x ) = ∑ t √ 2 π/ (2 l +1) f m l,t ( x ) r 0 ,b l,t . Finally, we can interpolate the reconstructed signal ˆ S on any shell-sampling D .

C.3 Details of compared models

We give an overview of details of the compared methods in Fig.13 and separate the different baseline methodologies into two groups.

Conventional Models. Conventional methods for fODF estimation generally solve the inverse problem iteratively. Constrained Spherical Deconvolution [66, 68] ( CSD ) and RUMBA [11] optimize voxel-wise dMRI reconstruction losses subject to a non-negativity constraint on the fODF. RUMBA-TV further enhances fODF smoothness by incorporating spatial total variation regularization. Importantly, conventional methods do not need prior training and are limited by the input data quality.

Deep-Learning Models. We evaluate our proposed framework against existing deep-learning methodologies for fODF estimation. While diverse network architectures, such as MLP, 2D/3D CNN, or spherical CNN, have been proposed, we adopt a standardized network architecture to focus our comparison on the inductive bias and training framework introduced by each method's convolution layer. We employ an MLP -based network that operates voxel-wise [52] and a CNN -based network operating on spatial patches [46]. Adding spherical inductive biases to the MLP network, we employ a voxel-wise S 2 -based network ESD [61, 19]. The SE ( 3 ) -equivariance PONITA method adds spatial information and inductive bias, equivariant to joint transformation on the spatio-spherical domain. Finally, the RT-ESD model is a E ( 3 ) × SO ( 3 ) -equivariance spatio-spherical convolution network, equivariant to independent transformation on the spatial and spherical domain. We extend RT-ESD to an efficient implementation SHD and spatially regularized SHD-TV .

C.4 Details of architecture and training

Network Details. A high-level network architecture is illustrated in Fig. 3. For E ( 3 ) × SO ( 3 ) -equivariant models, we use spatio-spherical pooling and unpooling operations. The SO ( 3 ) -equivariant models employ spherical pooling/unpooling, the CNN models use only spatial pool-

ing/unpooling, and no pooling is used by the MLP model. The first layer maps the input features to F s = 32 features for equivariant models and F s = 128 features for non-equivariant models.

The input and output feature number of the first and last layers depend on the convolution inductive bias, the number of estimated tissue components T , the number of input diffusion shells B , and the diffusion gradients angular resolution. The MLP and CNN models have at each voxel a feature vector of size F i = B × 45 (high angular resolution) or F i = B × 28 (low angular resolution) consisting of the 45 or 28 spherical harmonic coefficients of degree 8 or 6 , respectively. At each voxel, they output a feature vector of size F o = T × 45 , representing the 8 -degree spherical harmonic coefficients of the T tissue fODF. The S 2 U-Net, ESD , and RT-ESD models use an input spherical graph of 768 vertices (HEALPix grid resolution of 8 ) with F i = B spherical feature maps. The Concat-ESD model uses the same graph resolution but the number of feature maps increases to F i = B × 3 3 due to neighborhood concatenation. All four models output F o = T spherical maps (with the same resolution), corresponding to the T tissue fODFs.

SE(3)-

Equivariance

E(3)xSO(3)-

Equivariance

Spherical rotation-equivariant network

Standard networks

No Equivariance

Iterative optimization

Spatio-spherical equivariant networks

SO(3)-

Equivariance

T(3)

-Equivariance

Training implementation. All models are trained for 50 epochs with a batch size of 16 patches using the Adam optimizer [41] with an initial learning rate set at 1 . 7 × 10 -2 . The learning rate is decayed by a factor of ten after the 30 , 40 , and 45 epochs. For unsupervised models, we tuned the regularization weights using the ESD model on the DiSCo validation volume, and further tune λ tv for SHD : λ nn = 10 -1 , λ sparse = 5 × 10 -5 , and λ tv = 5 × 10 -1 . The spatially-informed models operate on 3 × 3 × 3 spatial patches, whereas the spherical-only models process 1 × 1 × 1 spatial patches.

fODF ground truth generation. Quantitative validation of the model requires access to ground truth fODF. The two benchmark datasets either do not directly provide this information or provide reference fODFs estimated with the CSD model on noise-free high-angular resolution dMRI volumes. However, both datasets provide ground-truth white matter streamlines. To avoid bias towards the CSD model and extract ground-truth fODF for the Tractometer dataset, we propose an alternative unbiased fODF estimation approach leveraging the ground-truth tractograph. We approximate the voxel-wise ground-truth fODFs by aggregating every streamline passing through a voxel and computing its spherical fiber density function. A white matter streamline can be represented as a set of vectors { v i j } where v ∈ S 2 is the local streamline direction, j is the streamline index and i is the vector index. For every voxel p , we find every local direction { v i k j k } going through the voxel. We then apply a spherical kernel density estimation on the set of unit vectors { v i k j k } using a uniform spherical kernel with an angular size of 15 ◦ . We then extract the ground-truth peak directions from the fiber density function using the MRtrix peak detection algorithm [68] with a maximum number of crossing fibers per voxel set to 10 .

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a convolutional neural network for spherical deconvolution of DWI data to estimate fiber orientation distribution. The main contributions over previous approaches include the introduction of Spatio-Hemispherical Equivariant Convolution, Dense Matrix Multiplication, and the use of Pre-computed Chebyshev Polynomials. These innovations improve overall efficiency. Evaluation was conducted on simulated data, assessing efficiency in terms of memory and runtime, and accuracy, demonstrating reduced false positive rates and angular error.\n\nSTRENGTHS:\n1. Enhanced efficiency for DWI spherical deconvolution using deep neural networks.\n2. Improved accuracy compared to current methods, demonstrated with simulated data designed for such experiments.\n3. Provided code for reproducibility.\n\nWEAKNESSES:\n1. Limited Novelty: The primary innovations over previous approaches focus on the network's efficiency (Section 3.2). Consequently, the main factors contributing to the reported improvement in accuracy remain unclear.\n2. Clarity of Presentation: The paper is highly detailed, making it difficult to follow and understand the main contributions that actually lead to improvements in performance (Section 3.2) and accuracy.\n3. Limited Demonstration of Impact: The practical impact beyond accuracy on simulated Tractometer data is not clearly demonstrated. It remains uncertain whether the proposed approach offers any significant clinical or scientific applications.\n4. Mixed Results: Figure 6D is particularly disappointing, as the results produced by the proposed method noticeably differ from the reference.\n\nQUESTIONS:\n1. Please clearly describe the main novel contributions of the paper and how they contributed to the results.\n2. Please thoroughly discuss Figure 6D, where the results of the proposed approach appear to deviate substantially from the reference.\n\nLIMITATIONS:\nThe authors discuss the limitations of the paper. However, the lack of demonstration of clinical applications and the generalization to multiple clinical/scientific DWI acquisition settings should be discussed more thoroughly.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors extend previous [work done](https://proceedings.mlr.press/v227/elaldi24a.html) in the diffusion MRI (dMRI) fibre orientation distribution function (fODF) domain with an efficient $\\mathbf{E}(\\mathbf{3}) \\times \\mathbf{SO}(\\mathbf{3})$ equivariant network. The proposed model directly leverages the antipodal symmetry of dMRI data to reduce computation time by 65%, as compared to previous work. The authors demonstrate the efficacy of this approach in a number of experiments; including an analysis of fODF estimation in simulated data and real world data, as well as in a downstream tractography task. They find that their method often performs best, whilst maintaining relatively high compute and memory efficiency.\n\nSTRENGTHS:\nThe writing and format of this work is excellent. The attention to detail, as provided within the main text and the appendix, rivals that of full length journal articles within this domain.\n\nWEAKNESSES:\nThis work represents a continuation of a previous method developed within [Elaldi et al](https://proceedings.mlr.press/v227/elaldi24a.html). Whilst the authors here present a significant increase in computational efficiency, this study is an iterative improvement on previous approaches, rather than a leap forward. I caveat that by acknowledging the importance of iterative improvements within scientific research.\n\nOverall, the writing is of an excellent standard. However, I have a small number of suggestions/mistakes enumerated below.\n\n- Line 98 you state that \"trainable models have the advantage of decreasing the reliability of the method...\". Here, _reliability_ evokes “reliable, as in you can count on it” rather than “rely on, as in this is a prerequisite”. I would maybe switch to “need for” or similar.\n- Line 151 I think you're missing a word at the end of the sentence \"sparse matrix multiplication significant computational...\"\n- Line 162 \"Fig. 3 overviews\", I would use \"presents an overview\" rather than using overview as a verb.\n- Line 269 I think \"unsupervisedly\" sounds a little clunky, would swap for \"in an unsupervised manner\" or similar.\n- Line 270 you state \"...to extract fODFs and then use the estimated fODF to investigate the effect of improved local fODF estimation on...\". I think this could be reworked to use the words fODF and estimat(ed/ion) a little less, perhaps by swapping \"the estimated fODF\" for \"them\", or swap \"investigate the effect of improved local fODF estimation\" for \"investigate their effect\".\n\nQUESTIONS:\nGiven that your experiments either involve simulated data or healthy patient data, when tasked with reconstructing fODFs for subjects with significant brain pathologies, would you reasonably expect to see a drop in performance? Or do you suspect that the regularisation enforced via the equivariant properties and loss functions would be enough such that the difference in performance would be minimal?More generally, how would you expect this method to perform when tasked with prediction on out-of-distribution data, as compared to the iterative per-subject CSD method?\n\nLIMITATIONS:\nThe authors have adequately addressed the limitations of their work", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work introduces a novel framework for fODF estimation through equivariant spatio-hemispherical networks that achieve dMRI deconvolution. Experiments on simulated dMRI datasets with known ground truth, as well as on real in vivo dMRI data are conducted, showing promising results while improving over previous methods.\n\nSTRENGTHS:\n1. The paper improves upon previous methods in both processing time, and quantitative results.\n2. The evaluation is sound and the experiments nicely show results on synthetic datasets with known ground truth.\n\nWEAKNESSES:\nThe main weakness of this manuscript is in the way the contributions section is written (at the end of the Introduction section, lines 57--71).\nThe authors would potentially increase readability of their paper by making this paragraph as clear and as sound as possible.\nIt would also help readers quickly identify if they wish to continue reading this paper and if it is of interest to their own research. \n\nMy suggestions are:\n1) Introduce this paragraph by restating what the main aims of this paper are (similar to what you wrote on lines 20--23). \n2) Clearly introduce the technical contributions as they are backed by the experiments / results section with a short description of what was achieved. \n3) Some of the contributions (specifically, the in vivo qualitative results) are not present in the main manuscript, but are part of the appendix. The exception is Figure 2 which does not have enough description in the main text (lines 254--256), and appears in the middle of a paragraph discussing the synthetic data results. I believe that the experiments section should be clearly reflected in the main aims and contributions of the paper, and the appendix should be used for optional / additional results which do not take away from the main contributions. I understand that there is a limit of 9 content pages to the paper, and I am happy to discuss this further.\n\nQUESTIONS:\nPlease find below some questions and general suggestions:\n\n1. Figure 1 introduces the readers to an example of how dMRI data looks, and it is an important prelude towards understanding the problem statement of your paper. For this reason, I suggest the authors include further explanations in this figure, in either visual form or in the captions, such as: \n    1. How does gradient 25 differ from gradient 288 (maybe try to explain / show that these are different gradient directions and/or strengths instead of the 1/25/288 indices which have no specific meaning in this context)? \n    2. I think it is also important to show a zoomed-in version of the T1w image, to not confuse the reader that the spatio-spherical signal is also present in the structural data. \n\n2. Please be consistent with referencing your figures in the manuscript: you sometimes write Figure x, and sometimes write Fig. x\n\n3. I suggest you introduce the name of your proposed spatial-hemispherical deconvolution (SHD) framework in the contributions section (lines 57--71) as on the next page Figure 2 shows examples of your model.\n\n4. In Figure 6B, could you discuss whether the low-resolution input to high-resolution output experiments could produce unrealistic reconstructions in the presence of noise / in a real dMRI data setting, as high-angular resolution is needed for higher contrast in the angular domain? I am wondering if for crossing fibers, for example, as shown in Figure 6D, none of the methods can accurately reconstruct the ground truth then maybe we cannot trust these reconstructions for low-angular resolution data? \n\n5. Can you also please label the x-axes in Figures 6A and 6B to make it clear that the values are in degrees?\n\n6. In Figure 6C can you explain the “narrowness” of your result as compared to the ground truth or CSD?\n\n7. Can you provide a short description (in section 4.2.1) of how the peak angular error and false positive rates are calculated? I understand that these are described in A.3, but to improve readability I suggest that they are introduced a bit sooner with the details left for the appendix, or at least to make it more clear that the details are in A.4. Moreover, it would be interesting to understand the slight increase in FPR in both Figures 6A and 6B when comparing SHD(-TV) with RT-ESD.\n\nLIMITATIONS:\nThe authors have attempted to address some potential limitations, but would be nice to see a lengthier discussion on how their proposed method would perform on other in vivo clinical datasets, under different noise levels, patient motion, etc.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel method for analyzing diffusion MRI data, leveraging convolutional network layers equivariant to the E(3)×SO(3) group, which respects the physical symmetries of dMRI data. The proposed spatio-hemispherical graph convolutions reduce computational complexity while maintaining high deconvolution accuracy.\n\nSTRENGTHS:\nThis paper presents a novel method for dMRI deconvolution by introducing equivariant convolutional network layers that account for the physical symmetries in dMRI data. The use of spatio-hemispherical graph convolutions, leveraging the antipodal symmetry of neuronal fibres, reduces computational complexity while maintaining accuracy. The proposed method addresses important challenges in dMRI analysis, focusing on the need for accurate deconvolution at clinically feasible resolutions.\n\nThe theorical foundation and empirical validation is sufficient, and the methodology is well-presented with clear explanations. The results are consistently validated, showcasing the method's efficiency and accuracy improvements. Additionally, The clarity of the paper is good with well-organized structure.\n\nWEAKNESSES:\n1. The reliance on specific assumptions may limit the scope of the model. It would be better to improve the flexibility of the model so that it could be applied to diverse scenarios.\n\n2. Diverse clinical conditions can be considered in future studies, involving varying levels of image quality and pathological changes. Conditions such as specific noise or artifacts are not fully explored.\n\nQUESTIONS:\n1. Can the authors clarify the limitations of the antipodal symmetry assumption? Are there specific scenarios where this assumption might not hold, and how might this impact the model's performance?\n\n2. Can the authors discuss the generalizability of their method across different clinical conditions and patient populations? How adaptable is the model to varying clinical data qualities?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-28"}