Equivariant spatio-hemispherical networks for diffusion MRI deconvolution
Authors: Axel Elaldi
Guido Gerig
Neel Dey
Venue: NeurIPS 2024 poster (2024)
Keywords: Geometric Deep Learning
Diffusion MRI
Spherical Networks
Biomedical Image Analysis
Primary area: machine_learning_for_healthcare
TL;DR: We create efficient equivariant networks for volumes where every voxel contains a spherical signal. We developed this method to improve fiber deconvolution in diffusion MRI, the primary method of mapping neuronal fibers in the brain."

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-hemispherica networks for diffusion MRI deconvolution
Axel Elaldi New York University axel. elaldi @nyu. edu
Guido Gerig New York University gerigenyu. edu
Neel Dey MIT CSAIL
dey@csail.m edu
Abstract
Each voxel in a diffusion MRI (dMRI) image contains a spherical signal corre- sponding 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) (O(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 accelerate the analysis of 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.
Introduction
Instead of scalar intensities, each voxel of diffusion MR image (dMRT) contains spatio-angular measurements of local water diffusion [64] As in Fig. [ this yields spatio-spherical images living on R3 x S2 that used to map neuronal organization vivo 7 44) alongside several other biomedical use cases (35 337 (6, (65) [71] Despite its potential, deriving neuronal fiber pathways using dMRI is hampered by significant partial voluming in both spatial and spherical domains due limited clinical scanner resolutions and low SNR [T. This paper presents a state-of-the-art dMRI deconvolution method recover neuronal pathways in practical timeframes by developing highly efficient equiv- ariant neural networks that account for dMRI's 's spatio-spherical data geometry and the voxel-level an- tipodal symmetry of neuronal fibers, all while demonstrating robustness clinical dMRI resolutions.
Need for deconvolution. The diffusivity at a voxel reflects the underlying local tissue microstruc- As multiple neuronal fibers recovering these fibers necessitates solving a blind fiber deconvolution problem at each voxel. 79 and this work focuses on the widely used fiber Orientation Distribution Function (FODF) model, which represents the fiber configuration voxel as sparse non-negative spherical function (66). rODFs
ture
within given resolution-limited voxel,
Several voxel-level fiber models have been proposed [5] 45 67 69
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., I 30 spherical samples per voxel), fODF recovery is highly ill- 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 nen-niiayyvyty constraints. Several extensions of CSD further regularize fODFs be sparse
and/or spatially smooth [2 29
Further, as fODFs can be assumed to be antipodally symmetric 40 67]. many iterative algorithms optimize only over the hemisphere accelerate
57
57
59 74
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Diffusion MRI Gradient 25
T1 MRI
Gradient
Gradient 288
Opatto-Sphertie Signal
Figure 1: A diffusion MRI (columns 1-3) and a Tlw MRI (column 5) derived from a subject in the HCP Young Adult dataset [70] The inset (column 4) visualizes region's spatio-spherical diffusion signal (b- 1000mm/ )²), highlighting crossing-fiber patterns and the grey/white matter interface.
ver-voxel optimization. For example, CSD only optimizes for the even-order spherical harmonics of the fODF and RUMBA [TT] uses approximately hemispherical sampling to represent the fODF. Despite significant progress using iterative solutions, these methods still require high angular clinically feasible resolve crossing fibers within voxel.
sampling resolutions that
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-trutt 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 -equivarrri 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-bue 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) x SO(3) -equivariant net- work 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-spherice modeling. However, its high computational requirements limit its use in clinical or large-scale deploy- ment. For instance, while subject-speciff iterative fits with CSD require 3-5 minutes for the whole brain, RT-ESD requires about a day
A100 GPU to converge when trained on a single subject.
Contributions. We present an efficient equivariant neural network framework for dMRI decon- volution that respects the data geometry of dMRI while ensuring computational practicality. Our vatio-hemissphricat 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 cumulative 65% reduction in processing time for an E(3) x -quivaraaa U-Net; (3) Finally, we use explicit mmoothhness-rromole spatial regularization, leading to further improved fODF recovery.
Experimentally, we achieve state-of-the-ar 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, the achieved efficiency gains enable training on a large set of human datasets, we can now train a single network for amortized inference new human dMRI, as opposed to the subject-specific optimization of RT-ESD. Our code is available at
com/ www.ellllii/eaeeeeeeeeeeelaaaaaa eeuiieriuuiiiiiaae -nni-decont
2 Related work
Deep learning for dMRI. Learned networks operating on dMRI have been extensively used to improve fiber tractography [46 633, super-resolve diffusion signals fODFs [48] 58 788 regress

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1. CSD
A.1) CSD
3-Sbell (igh-Angsie Resatutntt
1Shtel Low Angutrr Resoutton
BJ SHD-TV (ours) B-blcll High Asgulas Reselaton
B.1) SHD-TV cours) I Shell Low. Angular Resebution
Figure 2: A deconvolution visualization comparing recovered fiber orientation distribution functions (IODFs) produced by the widely-used iterative CSD [66] model (top row) and our proposed SHD- model (bottom row) with high-resolution / clinically-infeasibbl (left) and low-resolution clinically- feasible (right) spherical sampling. At sigh-resolutions (left), SHD-TV demonstrates enhanced localization of fiber orientations, heightened sensitivity to small- crossing fibers, and improved spatial consistency in the recovered fibers. At clinical low-resolutions (right), CSD struggles with the loss of input information, whereas our approach exhibits greater robustness to resolution losses and single-shell imaging protocols, yielding higher fidelity and spatially coherent fODFs. Appendix Fig. 9 visualizes comparisons with additional baselines.
microstructral indices in undersampled settings 13 27
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 RR$ x S images either through spherical networks applied voxel-
wise or approximate parameterizations of the spatio-spherical space 66
9
10 20
50 62]
For spherical data, (O(3) -qquivaraat convolutions are often achieved via spherical harmonics- parameterized 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. [0 develop convolutions for dMRI classification and super-resolution that exhibit voxel-wise SO -quiivranree and incorporate manifold valued spatial averaging. To model spatio-spherical dependence explicitly, SE(3) -eruvvv-rqvvaaeeeee for improved dMRI segmentation
47 at the cost of high-memory usage. In contrast, RT-ESD (0 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) (E(3)-equivari) -equivariant convolutions for spatio-spherical molecular graphs, but yet, does not scale to high-angular resolutions needed for dMRI deconvolution.
has been achieved via tensor field networks [50] and separable kernels [
Spherical deconvolution methods. Constrained spherical deconvolution (CSD) 38 66 and its sparse 112 and/or spatially regularized extensions [1] 29 57
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
59
43 46 51
52 61 22 78) that learn a mapping from diffusion signals to 1ODFs. 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 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
II] or fiber continuity
29 57
59
74), at the cost of increased optimization complexity. Current neural network frameworks
3

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have overlooked explicit fODF spatial regularization and rely solely on high-quality fODF training data and implicit spatial regularization by mean of aettial-eeigt sharing, either employing grid-wise 3D convolution 20
46
78
or channel-wise concatenation of neighhoring voxels 19 61 62
3 Methods
3.1 Background
dMRI deconvolution. A dMRI image is spatio-spherical function S R$ S + R with B features, called shells in the dMRI literature. The fODF model describes the tissue microstructure non-negative spatio-spherical function F, providing information the local tissue composition and orientations, and links the dMRI and the fODFs = C(F). dMRI deconvolution is interested in recovering the fODF from the dMRI by minimizing constrained reconstruction loss F argmin S C( We focus our study on learnable spatio-spherical deconvolution operators trained to reconstruct high-angular resolution fODFs from sparse dMRI measurements No(S)) = F.
Spatio-spherical convolutions. E(3) x SO(3) is the group of independent E(3) and SO(3) transformations, where E(3) acts on the spatial grid R33 and SO(3) acts on the voxel sphere S TR be learnable filter, such that convolving f and the yields four(T, ( RR) = S HER IpESs pESs the Vie(T"y R. pf. p) dy where (T, E E(3) x SO(3). RT- ESD [20] implements these convolutions by only considering isotropic filters to limit computational complexity. In addition, the filter the is expressed separable kernel the (2 22) øe, (21 (21)9o, (22), turning the convolution into a two-step spherical -spatial convolution:
S Let the R3
x
[ VERS
/ pES=
four q)
Φu, (1|= - yll)
00, (9,(9p" If p) dy
(f
We build on RT-ESD (00,, presented Fig. which uses DeepSphere [33 graph filtering based
80(3) -equivvrrr spherical convolution. At every spatial location y E R f(y..) .) is discretized on a set of spherical vertices = { E [eL,,.... [(y) E IR Nv. The graph G = (V, and its normalized Laplacian L are constructed from the set V, where E N the spherical adjacency matrix. The voxel-wise spherical filtering output then used to construct spatio-spherical graph fly) (L))()) .... TK (Lk-1(L)t (L)](y)] R with K features, where K the polynomial degree of the filtering and T (L) is the Chebyshev polynomial of degree k. A learnable isotropic
spatial filter ay -++ + then defined for each spherical filtered map fee resulting in the overall E(3) x SO 33(33)convolutiona operation expressed as:
(oue(T)
Σ MEP(2) R=
Σ
au (L)())))))) (L))(y).
(2)
where P(z) is spatial neighborhood around the position R Spherical (O (0(3)--quivariance 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 E V, A q) e lll-al//o with o's value selected to minimize empirical equivariance error.
3.2 Efficient dMRI spatio-hemispherical convolutions
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 & to hemispherical graph H. Because f is antipodal symmetric, we assume, without loss of generality, that a symmetric sampling, i.e. Vp V. 3(p)) E V. We construct the hemispherical sampling V+ {p E V and p. 2 0} from the spherical sampling V. with pa chosen to be the 3rd spatial coordinate of p. We compute the hemispherical Laplacian weight as,
P- L+(p,g) q) L(p. 4) + L(p.
Thus, the spherical operation Lf is entirely explained by the hemisphere graph Laplacian L- f+, ie. Vp V. 3(pt) E V+ s.t. (L* f)(=t) = (L)) where f is the f sampling on + (proof

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[A] Hemispherical Graph Construction Spherical Graph g
[B] IODF Estimation Framework
Hemispherical Graph H
Normalization
Reconstruction
Spatio Hemispherical dMRI
Vertieesp pu H
Reconsmmctin
Loos
Henisshle phere Reduction
Graph Laplacian L R RVv
Graph Laplacian L g
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Spansiy loss
V Vertices
V'Edges
1/2 V Vertices
1/4 Vddges
+ RALU PAAS caagicak
Spatial loss
Contribution overview. A. We reduce the spherical graph (g, L) an hemispherical graph (H, LT). B. The SHD deconvolution framework operates on a grid of spherical signals and reduces computation complexity while improving neuronal fiber deconvolution.
Figure 3:
in appendix A.1 For spherical samplings V such HEALPix, our proposed sampling reduces the number of vertices sampled to 3 V and the hemispherical Laplacian size to | Our reduction is both theoretically and empirically equivariant to E(3) x SO(3), see appendix A.2 for more details.
Dense Matrix Multiplication. DeepSphere's 53 used sparse matrix multiplication to compute T (L))(()) 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. SpepSp 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 7 (L)
(L)) = (2L7 (L) T (L))((z) for each new f and L. In our setting, we use the same spherical sampling 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 presents an overview of our proposed spatial- hemisphrimmipericcalll deconvolution (SHD) framework. Below, we first describe how 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 Sg sampled on 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 B0-diffusion signal is comparable between scans. Then, following [19], the normalized data is interpolated onto hemispherical HEALPix sampling V+, We provide details in appendix C.2
E(3) x (O Unne. The interpolated diffusion signal is then deconvolved to recover the spatio-spherical fODFs Fy+ on the same input HEALPix sampling V We use U-Net, with 4 depth levels, with layers acting spatial grid of hemispherical samples. We use our proposed layers the U-Net architecture presented in Fig. 3 Besides the last, every block comprises a convolution, batch-norm, and ReI.U activation. The last block consists of a convolution and Softplus activation function. The up/downsampling layers first involve mean spatial upsampling/poolinn on R followed by mean spherical upsampling/pooling on S², both introducing minor numerical cquivariance 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 Ng takes as input the shell-sampled dMRI signal Sg to produce fODFs F by reconstructing the dMRI signal 5% C(F - from the estimated fODFs, with details on the
(Sg)))) We train the network in an unsupervised manner
5

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Figure 4: Efficiency analysis. Runtime (A & C) and GPU memory usage (B & D) expressed as the percentage of the baseline (20). for both: (line plots) a convolutional layer applied to increasing angular resolution samplings and (bar plots) U-Net applied to high-angular resolution. The proposed convolution is more efficient than existing equivariant spatio-spherie convolutions.
reconstruction function C in appendix We note that the input and reconstruction shell samplings and DD need not be the same, allowing for angular super-resolution reconstruction during training required. The parameters 0 are estimated by minimizing a reconstruction loss subject to non-negativity and parosti--romoting regularization 19]:
19 66
2 F 1+
L= - 1 Ann. Aem.I|A.Fr |I 1
log
yt r²
112
+
AneLivi
where the terms correspond to reconstruction, non-negativity, sparsity, and smoothness, respectively. We define A as vector setting any positive component of Fy,, to O. We set 10 following [12] Lry is described below.
Spatial regularization. To add explicit spatial regularization, we propose to extend the spatial total- variation regularization from [1]] to the trainable model framework Ly = V y 113 where Vz the spatial gradient operator. The spatial regularizer L. 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 diversity of dMRI deconvolution settings on both real and synthetic datasets.
4.1 Runtime and memory efficiency improvements
Experimental details. We first compare the efficiency of our proposed convolution against two spatio- spherical 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 single apatio-sphericn 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 x 8 x 8 x V opatio-spherical volume, with V depending on the spherical HEALPix grid resolution. We the U-Net input HEALPix resolution to (V = 754) and we vary the single layer HEALPix resolution from (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.

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[A] Dataset and II: (ODP estimation
[B] Dataset MI: (ODF and Tracoogrphy Unsupervised fine-runing single volume
Generalization to volume
cMRI datimel
DISCO
Super-Angular fODF estimation
Unsupervised fODF estimation
Model
MMRI dessel
Model inpur
Exp, E
ODD
Reconstruction
Tractometer
low woslata
Exp, 2: Tractogrphy
Eqp. &
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EGp 2 Ire-rroniitann
[A] We perform super- resolved fODF estimation experiments on two datasets, DiSCo and HCP, respectively. Here, study the impact of using either high-angular or low-angular resolution as input. [B] We perform quantitative fODF and tractography estimation experiments on Tractometer. We extract fODFs and tractograms from the dMRI with both input and output having low-angular resolution.
Figure 5: Overview of the diffusion MRI experiments in Section 4.2
Results. Fig. 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 large spherical sampling. PONITA has a similar runtime and memory footprint as SHD on sparse spherical sampling but underperforms large spherical graphs (that common in dMRI deconvolution (12), being 10 times slower and more memory intensive. Consequently, a U-Net designed for high- angular resolution sampling is 2.5 and 3.5 faster using our proposed convolutions over RT-ESD and PONITA, while using and 20 times less memory, respectively. Our ablations indicate that using our hemispherical convolutions over the spherical convolutions, as well 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 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 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 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 single RTX8000 GPU and used less than 16GB of system RAM.
Baselines. We compare the proposed SHD method against conventional iterative methods including CSD (66). RUMBA and RUMBA-TV [I], and trate-of-the-ar learning based methods leveraging different convolutional layer, such as the non-equivariant voxel-wise MLP in [52) and the volume-wise CNN [46), that we also extend to have spatial regularization (C).-TV). We also compare against equivariant voxel-wise spherical convolutions ESD
61 19). and more recent equivariant spatio- spherical convolutions RT-ESD [20] and PONITA [ To focus on the inductive bias introduced by cach method's convolution layer, adopt 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- oon-angular resolution data, we further simulate low-angular
7

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[A] DiSCo IODF: High-angular resolution input
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Pasiivve Rute (
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Figure 6: DiSCo fiber detection performances on high (left col.) and low (right col.) angular resolutions. We first present fODF estimation results on high-angular [A] and low-angular resolution [B] imput (closer to bottom-left is better). [C-D] then present a qualitative example of two-crossing fiber estimation. Our faster implementation SHD does not negatively impact results in comparison to RT-ESD, while our improved model SHD-TV outperforms other methods by providing higher angular precision and less spurious fibers, especially at clinically-vabbe low-angular resolution.
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 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 5Ag First, we measure performance when all baselines are trained and tested on high- 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. O presents DiSCo deconvolution results for both the low and high- resolution settings. We visualize qualitative results on random HCP test subject in Fig[] 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 TODF estimation and tractography w/ known ground truth: Tractometer
A high-level overview of this experiment is described in Fig 5). As Tractometer consists of single volume with a low-angular resolution protocol with no held-out testing data, we benehmark methods on unsupervised fODF estimation similarly to previous work [9]. We first train the models in an
8

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[A] Tractometer: Unsupervised (ODF estimation
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SO@)-Equivaiinn
Not Expicssinnt
Conveniioaal
PUNTTA
0.50
RUMBA
5013) Equivoziazt
Equivarint
(ours)
R
0.45
RUMBA-TV
16
18
Angular 22
24
25
72
73
74
Valid Streamliee
26
28
79
[C] Tractometer: Downstream Tractography Brain-Stem Projection System
Ground- Truth
RUMBA
ESD
SHD (ours)
SHD-TV (ours)
Figure 7: Tractometer fODF estimation and tractography performance. Top: Unsupervised fODF estimation (A, closer to bottom left is better) and tractography (B, closer to top right is better) results. Bottom: In [C]. visualize ground-truth truth and estimated fibers projecting out from the brainstem into the right hemisphere. Overall, SHD and SHD-TV demonstrate more faithful fiber and streamline recovery as compared the voxel-wise RUMBA and ESD methods. In particular, SHD-TV yields fewer invalid streamlines and increases spatial coherence.
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 counted as Valid Bundles and the streamlines belonging one of the ground truth bundles counted as Valid Streamlines. Further, the volume of each estimated bundle is compared against the volume of its corresponding ground truth bundle, leading 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-qquivariant methods (CNN and CNN-TV) outperform most unregu- larized 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 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

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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 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 3.5x faster runtimes and up to 20x lower memory consumption as compared to existing spatio-spherica layers, while exceeding main- taining 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.
Acknowledgements
The authors gratefully acknowledge support from NIH grants IR01HD088125-01AL ROIHD055741, IRO1 18362-01, 2001EB021391-08 A1, RO1 ROIMHI and IR34DA050287.
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A Additional results
A.1 Spherical and hemisphere convolution
Hemispherical sampling. Consider the symmetric spherical sampling V = {p E 3²} such that Vp V. (-p) E V. and the Euclidean coordinate p (p2, Py.Ps) Ps) E S The sphere can be divided into a north and south hemisphere V = V UVV. with V = {p E V. such that ps > 0)} and V {p E V, such that p= 0)}. For point p lying
the circle p= = O we further add to V* every point such that Py > 0 and to V every point such that Py < 0. Finally, for the two points such that p= Py 0, we add p such that pz > 0 to V+ and p such that Pa to V-
Hemispherical restriction. Consider spherical function f S² - I and a function L S x S² 4 F. both sampled on V such that f E IRV and L E RVXV, Consider the function L+ S x S + R. (+ L(p q) sampling of f on V*
sampled on V+ such that L V/2x122 and ft E R V/2 the
Theorem Consider an antipodally symmetric spherical function f and L, such that for all p.q
V. f(p) = f(-p) and L(p,g) [(-p.-),, then vp E V, 3pt E V+, such that (Lf)(p)
(L (pt))
Proof Let p V. Then, define p+ if p V+, -p otherwise. Thus, we know that p+ E V+.
(If)(D) Σ ((p.))((())
(5)
BEV
Σ L(p,)))(()+ + Σ (pp
(6)
NEV+
QEV-
Σ L(p.))((g) + L(p, (9)((-))
SEVI
= Σ (L(p.g) + L(p. 99)){(9)
(8)
SEV+
Σ (L(p", .4) + L(p*,-))\(
(9)
SEv+
Σ (p=t,))(())
(10)
NEV+
(L* (p+)))
(11)
(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) x S0(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) x SO(3) convolution is compared to a grid-wise E(3)- equivariant convolution with spherical information, E(3)-SH, and voxel-wise -qquivvrrrr convolution with spatial information. Concat-SO(3) The E(3)- convolution is a 3D convolution with isotropic kernels processing spherical harmonic coefficients as input features [52.. The Concat- (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:
[[GN(]] - -((G))
TEquiv(N,G.,)
(13)
[[
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where N is the tested operator, f is spatio-spherical signal, and G = (T. E(3) x (O(3) composition of grid-wise and voxel-wise transformation. In practice we only test rotation- equivariance, and we limit 7 and R to be rotations. This quantity is measured over 1000 randomly sampled spatio-spherical signals f. random rotations (7, R), and randomly initialized W. f is randomly generated from uniform 2/(0, 1] distribution on 8 8 x 8 x V spatio-spherical volume with HEALPix hemispherical grid resolution 8 (V = 384 vectors). Moreover, we filter high spherical frequency by only keeping the first 8 spherical harmonic degrees.
Results. 8 illustrates estimated equivariance errors. As expected, we find that the E(3)-SH convolution exhibits lack of equivariance to voxel-wise rotations, and the Concat-SO convo- lution exhibits lack of equivariance to grid- wise rotations. The accelerated implementation of the (33) x (3(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
2,00
Type
Equivarlance Enro
1.50
1.25
0.50
0.25
0.00
Grad
Visel
Buth
Godd Rotation
Varel Rotatian
Buth Rouatian
Grd Rousion
Vocel Rosaion
Both
Rosation
Rosaion
E(3)x SO(3)
Concat-SO(3)
E(3)-SH
Convolution
Convolution
Convolution
Figure 8: Quantitative evaluation of equivariance error, depending on the convolution equivariance group, and the applied rotation group.
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 x 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 same synthetic generation process and have four BO images and four shells each with 90 gradients. We do not apply any pre-pree pre-processing step. For quantitative analysis, we train the different models on the first volume, validate the second, and test 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 2 tissue decomposition, and 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 [I7]: 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 Fl score. We report the angular error and false positive rate for the threshold that maximizes the Fl score the DiSCo validation volume. We report raw average scores and 95% confidence interval (CI) average over five random seeds in Table I Notice that all scores have 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
17

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Table DiSCo fiber detection performances noise level SNR=30 on high and low-resolution data. Models requiring training are trained on the first volume, validated on the second, and tested on the third. Results average over 5 random initialized models. Confidence interval at 95% given if CI
greater than 0.01.
Model
AUC (t)
FI
(1) Low
Angle @FI ())
FNR eFI (1)
FPR @FI
Protocol resolution
High
Low
High
High
Low
High
Low
High
Low
CSD
0.64
0.50
0.56
0.45
23.0
29.8
0.65
0.74
0.10
0.18
RUNBA
0.61
0.54
0.49
0.42
27.1
33.5
0.74
0.80
0.05
0.08
RUNBA-TV
0.49
0.55
0.49
0.48
26.4
27.5
0.71
0.74
0.12
0.09
XLP
0.39
0.48
0.53
0.42
23.8 = 0.1
31.8 + 0.2
0.68
0.78
0.12
0.15
CNN
0.56
0.51
0.51
0.47
24.9 + 0.3
27.4 1: 0.3
0.70
0.74
0.11
0.12
s" U-net
0.63
0.50
0.58
0.42
22.2 = 0.1
32.4 0.1
0.62
0.79
0.11
0.13
ESD
0.64
0.47
0.61
0.47
20.4 = 0.2
27.9 0.8
0.56
0,68
0.17
0.32
Concat-ESD
0.63
0.39
0.61
0.56
20.3 = 0.1
22.6 0.1
0.56
0.61
0.15
0.20
RT-ESD
0.64
0.60
0.60
0.57
20.4 =
22.0
0.58
0.61
0.14
0.16
SHD (ours)
0.64
0.61
0.60
0.57
20.0 = 0.1
21.5 0.5
0.57
0.60
0.17
0.19
SHD-TV (ours)
0.2
0.49
0.22
ensures the exclusion of twin subjects to prevent any data leakage between dataset splits. The dMRI volumes had undergone pre pre-proeesiig using FSL [36] and FreeSurfer [25]. and further details on the preprocessing pipeline can be found in the HCP dataset documentation 228 volumes adhered to high-resolution diffusion protocol, featuring 18 BO volumes per subject and three diffusion shells, each comprising 90 diffusion gradients. We apply similar downsampling the HCP dataset, randomly selecting 29 diffusion gradients from the lower b-value shell (1000, s/mm²) while retaining the BO 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-wir group-wise tissue response functions, accounting for white matter, gray matter, and cerebrospinal fluid present in in-vivo human brains.
The 100 dMRI
Validation details. In-vivo datasets lack microstructure ground truth. Some approaches 62 suggest using the CSD estimated fODEs 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. 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 its ability to discern the impact of rotation- equivariant methods compared to their non non-equivariant counterparts, Non -on-equivariant models may demonstrate high accuracy invariant scalar estimation while failing to capture the underlying geometric structure adequately. Moreover. PVE validation relies heavily on the accuracy of third- party tissue segmentation on T1/12 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 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 metries.
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A.1) CSD
A.11 CSD
3-hhell (igh-Angiar Rosotation
1-Sheli Low Angalar Resalatonn
B. ESD
B.1] ESD
Shell (lgh Angular Rosolation
S-hhil Low Angular Rosahitonn
Hi
C.1) CNN
C.2) CNN
Shell High Angular Rosolation
Shell Low Augular Resshitoo
D.1) SHD (ours) 3-bhell Augulag Resolation
Hrllerddy
D. SHD (ours) S-blll Low Augular Resolatuu
E. SHD-TV (ours) 3-bbill High-Aapula Howolatten
E.1) SHD-TV (ours) 1-blil Low-Aagllar Reselatas
Figure Qualitative illustration comparing the proposed equivariant dMRI deconvolution framework
against all other baselines.
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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 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 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 tumor border
Crossing- fiber area and tumor border
Tractography
Tractography
Figure 10: 1ODF and tractography estimation in a glioma-affecter brain (gray arrow). In [A]. we compare the conventional CSD method ([A.1]) with our proposed SHD-TV model ([A.2]). Our approach retrieves more spatially coherent fODFs with better fiber angular separation in voxels con- taining crossing fibers. Notably, the CSD method does detect fODFs in the crossing area indicated by the red arrow, leading to an inadequate representation of microstructures by the tractography algorithm. Additionally, our model does not reveal any abnormal fODEs near or within the tumorous tissue.
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B Equivariance vs. Data Augmentation: RS X S MNIST
Motivation. We study the advantages of using spatio-spherical inductive biases for convolutional layers, opposed to learning them from the data. We create synthetic spatio-spherical MNIST dataset (inspired by the spherical MNIST experiments of [13] to conduct controlled experiments.
Dataset. The R S MNIST data generation process is presented in Fig Importantly, the dataset characterizes spatio-spherical segmentation task, where voxel-wise information alone 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-lln fly data augmentation of spatio-spherical signals is computationally expensive, we pre-generate a total of 1000 volumes per dataset, with training-validattonn validation-test split of 716/142/142
Dataset generation. We randomly position eight oonrrleappping 4 x squares on 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 x 4 16 structures, termed tubes, aligned along the z-axis, each featuring the same classification digit across its voxels. Optionally, apply 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 projected onto sphere using the methodology presented in [T]]. We utilize HEALPix spherical sampling with a resolution of 4, equivalent to a spherical resolution of V = 192 vertices. reduce the amount
Given the straightforward nature of voxelwise spherical digit classification, of information at each voxel. Instead of projecting the entire MNIST image onto the sphere, we randomly crop it one-quarter of its original size before projecting the cropped digit onto the sphere. As result, any segmentation network has to rely on both the spatial neighborhood and the spherical information within voxel. Optionally, apply random voxel rotation the spherically projected digit.
A) Synthetic Volume Generation
B) Voxel-wise MNIST Spherical Projection
Cj R X MNIST
Label prneration
MNIST Random crop
Sploried Projection
Spherical MNIST
Grid
S Voxel Rodation
Rotation
D) Segmentation Dice Score (*)
E) Segmentation Dice Score (1))
No rolation during training
Random S rotation during training
Dice Store (%
0.76
Nesatetion ribation
Diee 5%
0.76
g² wibito
0.25
0.35
Conrat-80(3)
E(3)-8H
Cuncat-80(3)
E(5)-SH
EX()) 5O(3)
Convolatiio
Convalation
Convalation
Convolution
Convolution
Convelation
Convoiation
Convolution
Figure 11: Top row: Spatio-Sphericht MNIST generation process. Bottom row: Results from testing generalization from equivariance vs. data augmentation the synthetic R3 S MNIST classification task. [A-C] Dataset generation process. [D-E] Segmentation dice score of models trained without (D) and with (E) data augmentation. Overall, our proposed E(3) SO(3) convolution has a higher generalization power to unseen transformation than its non-equivariant counterparts, and the right inductive bias increases segmentation performance against data augmented- only models.
Model details. We compare four convolutional architectures, each exhibiting different equivariance properties while sharing the same U-Net structure presented in Fig We compare the three previously presented spatio-spherical E(3) x SO(3). E(3)-SH, and Concat-SO U-Nets, and, to highlight the importance of incorporating spatial information, we compare them with voxel-wise SO(3) U-Net. Inputs to the U-Nets consist of 16 x 16 16 192 g3 x S MNIST volumes, and the outputs,
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post- most-Softmax activation, are 16 x 16 x 16 x 10 volumes representing the classification probabilities of each voxel for the 10 digits.
Training details. Our training regimen uses a batch size of and trains for 50 epochs with an initial learning rate of 3 10-3, which halved after the 25, 35, and 45 epochs. The optimized objective function encompasses 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 first trained the dataset without data augmentation and subsequently tested on the dataset incorporating out-of-disributton random rotations. We then train and test the models the dataset augmented with random rotations. The results presented in Fig DI and H). To quantify segmentation performance, we compute the dice score across every volume in the testing
In Fig D), we observe that all four models, regardless of the embedded equivariance, perform well when trained and tested transformation-frre (i.c., 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 the rotation-free -fre 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 (3)
Fig 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 (O(3) models lacking spherical or spatial inductive bias improves segmentation performance but falls short of matching the performance E(3) x SO(3) equivariant model.
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C Additional experimental details
C.1 Spatio-Hemisshericall Convolution overview
A) Input Neighborbood
B) Vaxel wise Hemispherical Filtering
Filtered Neighborhood
((if(())
Graph
[(e) e 2
Fie) ta
bemisppatical verticn
D) Vertex Weight Sharing
E) Vertex--wise Spatial Convolution Isotropic Weights 3 P
F) Convolution Output
spectal
F. sippul, Glierd,, oslpul signais
((s) sphectcal digual
Kuta spottal signal
Convolution nilli sllD//a)
laglacin bemisphaetcal graph
24 clebbysher polpuctuaal
P:R g
fas Par
K polyuuiial degres
o'Jrrnispaecical vertox
feap R2 R
Fu. Fas laput and outgut feadrurve
Figure 12: Overview of the Hemispherícal E(3) SO(3) Convolution computation. This figure is adapted from (20]. [A-C] The input of the convolution is a 3D grid of hemispherical graphs with V vertices per voxel. The input is first processed by voxel-wise spherical filtering using the proposed hemispherical Laplacian and efficient implementation. [D-F] The 3D volume is then processed by 3D isotropic convolution with weight-sharing across the hemispherical graph vertex.
C.2 Details of the deconvolution framework
IODF model. The fODF model assumes the voxel-wise diffusion signal to be produced by a composition of tissue types, with each tissue further composed of 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 [= are tissue-dependent antipodally symmetric spatio-spherical functions. The response functions RF=(REE)) are tissue- desendent and shell- dependent spherical functions, assumed constant across the spatial domain, and to be antipodal and axial symmetrie. The total i-shell dMRI signal is modeled as a sum over tissue-specific signals 5h = Et= S where tissue-specific diffusion signals are modeled as a voxel-wise spherical convolution St = C(F,, RE between the fODF and the response function. The convolution is defined as C(F, (x, Spe.s F(x. p) RF( p)dp.
Diffusion sampling. A dMRI image S is a function S(z, R x S + R. In practice, the dMRI signal is discretized on that for every E B, the b-shell function Sh a spatio-spherical signal. Furthermore, the b-shell func- tion is sampled on a shell-dependent set of spherical gradient direction V/, {g E S 82) ]e[L,..NN) with N₂ the number of b-shell gradient sampling. We note Sb.vo. (x) [S(x, o. b), b)..... S(x, L3, R.V, the sampled b-shell signal on Vg. Moreover, we note
shells B = (b, E R ]re[L...B] where B is the number of shells, such
{B, (V) (eB) } the full diffusion sampling set, and (D) = [Sb, Vu, (a),..., S Vog (a:) E NR the sampled dMRI signal, with No Ebeg No the total number of gradient sampling.
Data normalization. For every new dMRI scan Sort, we compute the scan-specific white matter BO response function Or.. uim using MRuii (68), and normalize the dMRI signal 5 Sori. / The input of our framework is the normalized shell-sampled diffusion MRI signal Sz(x)
[Sb,. Sp.. Von (x)] E IR No for x E R²3 Following [19], we first normalize the protocol- dependent 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 Sp (zx R2 for every
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b E B shell, where Ny is the number of harmonic coefficient for a maximal harmonic bandwidth of the odd-order spherical harmonic coefficients are equal to O and we only compute the even-order coefficients, making the total number of coefficient
L. Because the diffusion signal is
Nr = (I/2+1)(L 11. Importantly, getting the coefficients (1) from Sb, vg (I) requires at least ([(/2+1)(L+)) samples. We limit the maximum interpolation bandwidth to L = 8, which requires at least 45 samples per shell. In case the shell sampling V. 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+, Sb (æ) = Sb(2) Y 5 where Y V+ relates the even L bandwidth harmonic coefficients to the HEALPix spherical sampling. We get the B feature input spatio-spherical signal Sy- (ar) E RVxB
2
Group-average response function. We estimate a population aapulation-based response function RF following [55]. For N₂ normalized dMRI scans in our training dataset, we first compute ' tissue response functions RF,,; f,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 RF, = 1/N, Σ N, RF,,-
From the fODFs Fy-, we reconstruct the diffusion signal Sp the reconstruction shell-sampling D using a spherical convolution with response functions R.F. For every voxel x E R3, we compute the spherical harmonic coefficients F(x) (f]](z)} E N from the estimated F (z). The fODFs being antipodal-symmettie we only compute the even-degree coefficients. The maximum spherical harmonic degree L depends on the resolution of the HEALPix sampling V+, We use sampling resolution with 384 vertices per hemisphere and a fODF maximum spherical harmonic degree of T, = 18. Moreover, the response functions are also represented by their spherical harmonic coefficients R'F = (ritee) E xT'xx The RFs are antipodal and z-axis symmetric, setting to every non-even degree and non-zero order coefficients. We then compute the reconstruction dMRI spherical harmonic coefficients Š(x) = {s/ (s/">(a)) (x)} E N $1 m.hh (xx = E. 2m/ + (z)r) Finally, we can interpolate the reconstructed signal $ on any shell- selllsan D.
Signal reconstruction.
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 [ optimize voxel-wise dMRI reconstruction losses subject to a non-negativiyy constraint 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 s convolution layer. We employ an bbP-b network that operates voxel-wise [52) and CNN-based network operating on spatial patches [46]. Adding spherical inductive biases to the MLP network, we employ a voxel-wise ba-ed network ESD 61 19]. The SE(3) (3)))vuivarii PONITA method adds spatial information and inductive bias. equivariant to joint transformation the spatio-spheriin domain. Finally, the RT-ESD model is a E(3) x SO(3) (e)))v-raaaa spatio-spherical convolution network, equivariant 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) S0 equivariant models, we use spatio-spherical pooling and unpooling operations. The (0 equivariant models employ spherical pooling/unpooling, the CNN models use only spatial pool-
24

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ing/unpooling, and no pooling is used by the MLP model. The first layer maps the input features to F. 32 features for equivariant models and F,- 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 feature vector of size F. the 45 or 28 spherical harmonic coefficients of degree 8 or 6, respectively. At each voxel, they output feature vector of size Fo T x 45, representing the 8-degree spherical harmonic coefficients of the T tissue fODF. The S U-Net, ESD, and RT RSD models use an input spherical graph of 768 vertices (HEALPix grid resolution of 8) with F. B spherical feature maps. The Concat model uses the same graph resolution but the number of feature maps increases to F; = B x 33 due to neighborhood concatenation. All four models output F. = 7 spherical maps (with the same resolution), corresponding to the T tissue fODEs.
45 (high angular resolution) or F; B x 28 (low angular resolution) consisting of
Iterative optimization
Spherical rotation-equivariann network
CSD (Tourniee et al. 2007)
BSD (Elaldi et al., 2021)
S0(3) Equivasianee
S00))-Eqivataane
RUMBA (Canalos-Roxdrigues et al., 2015)
RUMBA-TV (Canales- (Canalos-R<drigues et al., 2015)
Spatio-spherical equivariant networks
Standard networks
PONITA (Bokkers 2024)
SE()-Equivariaee
XLP (Nath et sl., 2019)
Equivariance
RT E3D (Elaldi et al., 2023)
cXn (Lin al., 2019)
SHID (Ours)
B
T(3) Equivariance
CNN TV (ablation)
SID-TVV -V- (Ours)
Model description. We compare conventional and learnable methods. Learnable methods have increasing embedded geometric prior through equivariance.
Figure 13:
Training implementation. All models are trained for 50 epochs with a batch size of 16 patches using the Adam optimizer [4]] with an initial learning rate set at 1.7 x 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 Ato for SHD: Ann 10-1, Asparse = 5 x 10-5, and Ato = 5 x 10-1 The spatially-informed models operate on 3 x 3 spatial patches, whereas the spherical-only models process 1 x 1 spatial patches.
TODF ground truth generation. Quantitative validation of the model requires access to ground truth TODF, The two benchmark datasets either do not directly provide this information provide reference fODFss 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-tre -tuth 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 where 2 is the local streamline direction, j the streamline index and is the vector index. For every voxel p. we find every local direction {uf: going through the voxel. We then apply spherical kernel density estimation on the set of unit vectors {vft) } 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 maximum number of crossing fibers per voxel set to 10.
of vectors {vf)
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Reviews:
[{"review_id": "i92l68BfO8", "reviewer": "Reviewer_JrFU", "summary": "This 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.", "questions": "1. 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?", "limitations": "Yes.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This 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.", "weaknesses": "1. 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the positive evaluation and for highlighting the importance and quality of our methodology, presentation, and experiments. \n\n# Antipodal symmetry assumption\n\n> _\"It would be better to improve the flexibility of the model so that it could be applied to diverse scenarios.\"_\n\nWe agree that the antipodal spherical symmetry assumption limits our approach to antipodally symmetric spatio-spherical data. While previous methods \\[[1](https://arxiv.org/pdf/2310.02970),[2](https://arxiv.org/pdf/2304.06103),[3](https://arxiv.org/pdf/2102.06942)\\], have developed spatio-spherical equivariant networks without this assumption, they are far too computationally intensive to scalably process high-dimensional data such as diffusion MRI. We believe that this is partly why the diffusion MRI community primarily still uses conventional iterative methods.\n\nWe instead build on previous works by incorporating more knowledge of the underlying task directly into the network layers to significantly increase computational efficiency. However, we agree with you that future work should aim to relax these assumptions while maintaining our efficiency gains and this is mentioned in our future work.\n\n> _\"Can the authors clarify the limitations of the antipodal symmetry assumption?\"_\n\nThank you for initiating this important discussion! The antipodal symmetry assumption is widespread in dMRI analysis due to the symmetric nature of the diffusion process. Consequently, only a symmetric fODF can be estimated from a single voxel-wise diffusion signal, and **mainstream deconvolution methods widely use this assumption**. For instance, the most widely-used fODF estimation method, Constrained Spherical Deconvolution \\[[4](https://www.sciencedirect.com/science/article/abs/pii/S1053811907001243)\\], computes only the even-order spherical harmonics of the fODF, thus making the antipodal symmetry assumption.\n\nThe one limitation is that at a _microscopic_ level, fODFs can have antipodal asymmetry. However, this is only visible in ex vivo dissection studies and we focus on in vivo clinical dMRI where antipodal symmetry is a valid assumption. This assumption is also reflected in common dMRI acquisition strategies in large-scale studies that only sample hemispherical signals at every voxel \\[[5](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8317510/),[6](https://onlinelibrary.wiley.com/doi/10.1002/mrm.21646),[7](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7065087/)\\]. We will mention this in the Discussion of the revised paper.\n\n# More diverse clinical conditions\n\n> _\"Diverse clinical conditions can be considered in future studies, involving varying levels of image quality and pathological changes [...].\"_\n\nThank you for suggesting this possible future direction! We do agree that our model can be further validated under more diverse clinical conditions. However, as there is no _in-vivo_ microstructural ground truth for the human brain, validating novel methods is challenging and ill-posed. \n\nIn our paper, we rely on community-developed standardized benchmarks and datasets. As mentioned by the reviewer, our paper presents comprehensive experiments to quantitatively validate the performance improvements of our method by performing:\n- fODF estimation analyses with diverse image qualities, quantitatively on the DiSCo and Tractometer datasets, as well as qualitatively on the HCP dataset.\n- Downstream tractography task analyses on the Tractometer dataset.\n- Speed and memory analyses.\n\nRegarding the specific scenarios mentioned by the reviewer,\n\n## Brains with lesions\nIn Fig. 1 of the rebuttal PDF, we perform an initial fODF analysis on brains with anomalies using the recently released dataset from \\[[8](https://www.nature.com/articles/s41597-024-03013-9)\\]. 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.\n\nHowever, we note that this analysis is preliminary and that tractography on brains with lesions is a highly active area of research that requires substantial modifications to tractography algorithms \\[[9](https://www.sciencedirect.com/science/article/pii/S1053811921009241)\\], which are outside the scope of our fODF estimation work. We will mention this as a limitation and area for future work and add these new results to the appendix.\n\n## Varying image qualities\n\nRegarding image qualities, all of the datasets in our paper have substantially different characteristics and SNRs. Further, we believe our model generalizes well across **different clinical conditions and patient populations** because:\n- It is trained to reconstruct the input diffusion signal directly, rather than recognizing specific tissue structures that may or may not be present at clinical deployment. \n- The self-supervised training framework allows the model to be fine-tuned for any new image and imaging setup. \n\nWe agree that testing performance under varying amounts of patient motion and imaging artifacts would be highly interesting. However, we are not aware of any publicly available in vivo datasets that provide such data. Our limitations and future work will now further emphasize that future work should investigate performance under various amounts of head motion or other clinical artifacts.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qrHyWyv39H", "reviewer": "Reviewer_zoNj", "summary": "This 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.", "questions": "Please 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.", "limitations": "The 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.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. 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.", "weaknesses": "The 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the highly detailed and valuable feedback! We will incorporate the suggestions and address the other concerns below:\n\n# Weaknesses\n\n> _\"The main weakness of this manuscript is in the way the contributions section is written at the end of the Introduction. [...]\"_\n\nThank you for the detailed suggestions on editing this paragraph! We will incorporate them into the revised paper.\n\n> _\"Some of the contributions (specifically, the in vivo qualitative results) are not present in the main manuscript, but are part of the appendix.\"_\n\nAs the reviewer mentions, we combined the qualitative in-vivo fODF estimation and quantitative synthetic analysis in Section 4.2.1, as both address the same question. We clarify that this section details both results and is not solely focused on quantitative outcomes. \n\nHowever, we agree that our complete qualitative and quantitative analysis was not entirely contained in the main paper due to space limitations. In our revision, we will expand the caption of Fig. 2 and bring in more qualitative in vivo results into the main paper from the appendix. To do so, we will abbreviate the front matter of the introduction and move some lower level experimental details to the appendix.\n\n# Questions\n\n> _\"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 low-resolution dMRI data setting [...]?\"_\n\nThank you for raising this important discussion point. We agree with the reviewer that hallucination is a concern in all undersampled reconstruction methods, including previously proposed fODF estimation methods. \n\nAs demonstrated in our experiments, in low angular settings, the current widely-used Constrained Spherical Deconvolution (CSD) method provides inaccurate fiber estimates with high angular error. Our motivation is precisely to mitigate these unrealistic reconstructions by considering the spatial correlation of the diffusion signal and leveraging the network's equivariance property. \n\nIn our DiSCo experiments with clinically relevant noise, as shown in Fig.6.B, using both the spatially-informed network and spatio-spherical equivariance leads to a significant decrease in hallucinated false positive fibers and angular error. **Importantly, the FPR achieved with our method on low-angular resolution input is competitive with the results on high-angular resolution input**.\n\nHowever, we entirely agree that undersampled reconstruction results must be approached with caution, as the lower angular resolution increases angular error and the likelihood of missing estimated fibers, which can be qualitatively observed in Fig.2.B. This caution should be considered within the broader context of the tissue microstructure estimation field. For example, Diffusion Tensor Imaging, which uses only a few spherical samples to estimate the underlying fiber configuration, can sometimes create spurious local reconstructions but remains a primary tool in clinical applications and research. Our method, with its additional priors and self-supervised learning strategy, represents a significant improvement over methods currently used in clinical settings.\n\nWe will clarify the results to include this discussion.\n\n> _\"[...] lengthier discussion on how the proposed method would perform on other in vivo clinical datasets, under different noise levels, motion, etc.\"_\n\nWe agree that this is an important discussion. In the attached rebuttal PDF, we perform an additional experiment using dMRI images of brains with pathologies \\[[1](https://www.nature.com/articles/s41597-024-03013-9)\\]. As also suggested by reviewer `JrFU`, we agree that our method would benefit from further specific validation under varying image qualities and the presence of artifacts.\n\nBriefly, we find that additional priors of spatio-spherical equivariance and spatial smoothness improve fiber estimation on in vivo anomalous clinical data, with the caveat that such analyses need to be performed on much wider scale in a clinical followup for certainty. Further, we believe that as image quality degrades, that is precisely where additional priors help for ill-posed reconstruction. Due to space limitations in the rebuttal, please see our discussion with Reviewers `JrFU` and `F5CU` for further details. These limitations and avenues for further work will be added to the revision.\n\n> _\"[...] slight increase in FPR in both Fig 6A and 6B when comparing SHD(-TV) with RT-ESD.\"_\n\nThe reviewer is correct that there is a slight increase in the FPR. \nRegarding the SHD versus RT-ESD comparison, we speculate that as FPR and Angular Error have a tradeoff, they might require slightly differing regularization weights. SHD also has the prior of antipodal symmetry whereas RT-ESD does not, and the bias-variance tradeoff might be causing this slight increase. \nFor the proposed SHD-TV model, the increase in FPR can be attributed to the smoothing effect of the total variation regularization, which occasionally extends a fiber into neighboring voxels where it might not be appropriate. However, this increase in FPR is offset by the benefits of much lower angular error and higher spatial coherence, which enhances robustness against noise and improves fiber localization.\n\n> _\"In Fig6C can you explain the “narrowness” of your result as compared to the ground truth or CSD?\"_\n\nWe use a prior of sparse fibers using a sparse regularizer during training, this is consistent with previous work such as ESD and RT-ESD and the wider fODF literature. This leads to increased narrowness in comparison to the CSD method, which does not use any sparsity regularization.\n\n# Writing improvement suggestions\n \nThanks again for all of the detailed writing and presentation suggestions. We will incorporate all of them into the revision.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RUfDsmCZHN", "reviewer": "Reviewer_F5CU", "summary": "The 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.", "questions": "Given 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?", "limitations": "The authors have adequately addressed the limitations of their work", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The 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.", "weaknesses": "This 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\".", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the encouraging feedback! Broadly, we will incorporate all of the detailed suggestions and address the remaining high level questions/comments below:\n\n> _”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\nWe agree that we are building extensively on previous work, in particular the spatio-spherically equivariant RT-ESD method. However, RT-ESD is too computationally intensive for scalable use in either research or the clinic. \n\nAs a result, we develop a principled framework that leverages a physically motivated assumption about dMRI to build new equivariant layers that retain the equivariance, but are much faster to compute and store in memory. Further, we conducted an extensive analysis of these network kernels and identified key avenues for improvement (eg, pre-computing the Chebyshev polynomials) in order to make deep learning on very high dimensional dMRI data tractable. \n\nWe agree that this is not necessarily a paradigm shift methodologically, however, our contributions have significant potential for real-world application which was not possible with previous work.\n\n> _\"I have a small number of suggestions/mistakes enumerated below.\"_\n\nWe greatly appreciate the detailed enumeration of typos and awkward phrasings! We will make all suggested changes.\n\n> _\"[...] When reconstructing fODFs for subjects with significant brain pathologies, would you reasonably expect to see a drop in performance? [...] how would you expect this method to perform on out-of-distribution data [...]?\"_\n\nThank you for raising this question. We do agree that our model can be further validated under more diverse clinical conditions. As there is no _in-vivo_ microstructural ground truth for the human brain, especially for brains with pathologies, validating novel methods is challenging and ill-posed. \n\nHowever, as testing data distributions shift, we believe our additional proposed priors benefit ill-posed reconstruction. Our priors of spatial smoothness, spatio-spherical equivariance, sparse fibers, and antipodally symmetric signals are all valid for brains with pathologies as well. We therefore do not expect degradation on brains with lesions.\n\nAs a preliminary exploration, in Fig. 1 of the rebuttal PDF, we now perform an initial fODF analysis on a brain with a tumor using the recently released dataset from \\[[1](https://www.nature.com/articles/s41597-024-03013-9)\\]. We find that the additional priors from our method help fODF estimation and fiber tracking substantially yielding smooth fibers, whereas the baseline CSD method has a hole in its estimated fiber tracks.\n\nHowever, we emphasize that this analysis is preliminary and that tractography on brains with lesions is a highly active area of research that requires substantial modifications to tractography algorithms \\[[2](https://www.sciencedirect.com/science/article/pii/S1053811921009241)\\], which are outside the scope of our fODF estimation work. We will mention this as a limitation and area for future work and add these new results to the appendix.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dCXhvPq5KR", "reviewer": "Reviewer_rCgS", "summary": "The 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.", "questions": "1. 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.", "limitations": "The 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.", "rating": 3, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. 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.", "weaknesses": "1. 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the valuable feedback and for highlighting our improved efficiency and accuracy in comparison to current work! \n\nWe believe that there may have been some miscommunications on our part and hope to clarify them below:\n\n# Unclear reasons for improvement\n> _\"Limited novelty as presentation focused on efficiency. The main factors contributing to the reported improvement in accuracy remain unclear.\"_\n\nTo clarify, the accuracy gains primarily come from using physically motivated priors to solve highly ill-posed fODF reconstruction problems.\n\nTo recap, previous work either\n- Does **voxel-by-voxel** iterative optimization (CSD) or unsupervised equivariant network training (ESD), accounting for only the spherical nature of diffusion signals.\n- More recent work (RT-ESD) does unsupervised equivariant network training accounting for both spatial and spherical symmetries. However, it is **too computationally intense** to run practically, which is half of our motivation.\n\nOur accuracy gains come from two perspectives,\n1. We now correctly model the **antipodal symmetry** of diffusion signals, broadly seen in clinical in vivo dMRI, and directly incorporate it into our network kernels. This allows the network to focus on the reconstruction task instead of also trying to learn the antipodal symmetry of the data. This convolution is detailed in Section 3.2/”Spatio-hemispherical equivariant convolution”.\n2. Further, we incorporate a physically-motived regularizer encouraging low **spatial total variation** into the unsupervised network training. To our knowledge, this regularizer has not been used in previous deep dMRI deconvolution networks. Its inclusion significantly improves fiber localization over previous methods, particularly when working with clinically used low-angular resolution images.\n\nBoth contributions, alongside the low-level network kernel analyses, are entirely novel and contribute to the observed efficiency and accuracy gains.\n\nWe would be happy to provide any additional information on these contributions will directly emphasize these perspectives in the revision.\n\n\n> _\"The paper is highly detailed, making it difficult to follow and understand the main contributions that actually lead to improvements in performance and accuracy.\"_\n\nThank you for raising this concern. We have outlined the main contributions that led to the reported gains in the answer above.\n\nRegarding the contribution presentation, we will incorporate your feedback (and that of Reviewer `zoNj`) to revise the presentation to be easier to follow, thank you for raising this concern. \n\nFor reference, the information regarding the causes of performance increases are currently presented in the following sections:\n- The theoretical and technical explanation of the improved efficiency is provided in Section 3.2, with a quantitative analysis of each contribution detailed in Section 4.1 and illustrated in Fig. 4.\n- Section 3.3 covers the contributions to improved fODF estimation accuracy. A detailed quantitative analysis of our method is presented in Section 4.2 and depicted in Figs. 6 and 7, emphasizing the importance of total variation regularization during training.\n\nWe will streamline this in the revision.\n\n# Experiments\n> _\"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.\"_\n\nWe respectfully disagree. Our experiments include quantitative analyses on two benchmark datasets widely used by the dMRI community and also include qualitative analyses on in vivo HCP data, widely used in a variety of dMRI papers, including at NeurIPS \\[[1](https://arxiv.org/pdf/2011.01355),[2](https://arxiv.org/pdf/2306.00854)\\].\n\nFor context, fiber orientations in human diffusion MRI at clinical resolution have no ground truth. Acquiring this ground truth would require post-mortem dissection-based analyses and we are not aware of any publicly available dMRI dataset of this kind. As a result, the dMRI analysis community focuses on highly realistic simulated benchmark datasets to measure algorithmic progress. This is consistent with the challenge datasets of DiSCo and Tractometer used in our paper, where we achieve state-of-the-art fODF reconstruction results at more practical speeds.\n\nFurther, our method is entirely generic and can be plugged into any existing dMRI analysis pipeline for analyzing the connectivity of the human brain. Lastly, the most accurate of previous work (RT-ESD) was unsuitable for clinical and scientific applications due to its computational load. Instead, our work is highly scalable due to its efficiency gains and can be practically deployed and matches or exceeds its accuracy.\n\nWe would welcome any further discussion on this matter and will clarify the text to emphasize these points.\n\n> _\"Fig6D is disappointing, results produced by the method noticeably differ from the reference\"_\n\nTo clarify, Fig. 6D is a qualitative result of a single deconvolved voxel at clinical resolution. The actual **dataset-wide results are presented in Fig. 6B**, wherein we achieve much fewer false positives with lower angular error when looking at the entire dataset.\n\nFor context, this is a highly undersampled dataset that is challenging for all baselines and we cannot expect reconstructions from low-angular resolution to match the performance on high-angular resolution. However, the baseline CSD produces an entirely different fiber orientation and ESD and CNN produce spurious fibers. Only SHD-TV has the correct number of major fibers with a lower angular error to the ground truth.\n\nAgain, these are highly undersampled inputs so no method will achieve perfect reconstruction due to the ill-posedness, but dataset-wide, we measure a 30% decrease in angular error, and a 27% decrease in missing estimated fibers (false negative rate), when using SHD-TV in comparison to CSD, for example.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank the reviewers for their time and their encouraging feedback. Their comments and suggestions have made the revision a stronger paper.

We were happy to find that they found the submission to be theoretically sufficient \[`JrFU`\], well presented and organized \[`JrFU, F5CU`\], extensively validated with sound evaluation \[`JrFU`, `zoNj`\], yielding better efficiency and accuracy \[`JrFU`,`F5CU`,`rCgS`\], with a high attention to detail \[`F5CU`\]. 

Common questions and concerns revolve around the clarity of our contributions paragraph in the Introduction \[`zoNj`, `rCgS`\], our antipodal symmetry assumption \[`JrFU`\], the limited analysis in the main text of the in-vivo experiments \[`zoNj`\], and a lack of clinical applications \[`rCgS`\]. The reviewers would also like to see a longer discussion of our method in more diverse clinical settings, such as pathological changes, noise, and motion  \[`JrFU`, `zoNj`, `F5CU`\].

## Changes and clarifications

To improve our submission and address the reviewers' concerns, we make the following major revisions and clarifications, briefly recapped below:
-  \[`JrFU`\] We motivate the physical reasons behind the assumption of antipodal symmetry of diffusion MRI signals and describe the computational benefits below.
-  \[`zoNj`,`rCgS`\] We will use the suggestions to rewrite our contribution paragraph in the revision so as to make the paper accessible and clearly understandable.
-  \[`zoNj`,`rCgS`\] We clarify our experiments on the highly undersampled qualitative example visualized in Fig 6D.
-  \[`zoNj`,`rCgS`\] We provide additional descriptions of our quantitative and qualitative results, regarding the narrowness of the predictions, slight increases in false positive fibers on a specific experiment, and the challenges of prediction on low-angular resolution undersampled images.
-  \[`JrFU`,`zoNj`,`F5CU`\] We now provide a qualitative analysis of the outcome of our method on dMRI of a brain with pathologies and discuss how our model adapts to new clinical and imaging setups. 

All details and remaining comments are addressed in the individual responses below. Additional results are attached as a PDF.

We appreciate the reviewers' comments on our contribution and thank them again for their time and expert feedback. We would be happy to address any additional concerns and incorporate any further suggestions.

Meta-review: Two reviewers recommend Weak Accept, another reviewer recommends Accept and a fourth reviewer recommends Reject. The main argument of reviewer rCgS to recommend Reject is that the subject of the paper is better suited for a "dMRI / Neuroimaging-related conference than to the broader audience in neurips". The authors rightly provide a battery of good arguments as to why a paper such as this has a place at NeurIPS. I fully agree with the authors. Furthermore, the paper goes beyond the simple application of a method to the dMRI domain and proposes a new equivariant deep-learning architecture which is particularly designed for the peculiarities of dMRI data. I therefore recommend Accept.
Decision: Accept (poster)