Thank you for the valuable feedback and for highlighting our improved efficiency and accuracy in comparison to current work!

We believe that there may have been some miscommunications on our part and hope to clarify them below:

Unclear reasons for improvement

"Limited novelty as presentation focused on efficiency. The main factors contributing to the reported improvement in accuracy remain unclear."

To clarify, the accuracy gains primarily come from using physically motivated priors to solve highly ill-posed fODF reconstruction problems.

To recap, previous work either

• Does voxel-by-voxel iterative optimization (CSD) or unsupervised equivariant network training (ESD), accounting for only the spherical nature of diffusion signals.
• 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.

Our accuracy gains come from two perspectives,

1. 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”.
2. 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.

Both contributions, alongside the low-level network kernel analyses, are entirely novel and contribute to the observed efficiency and accuracy gains.

We would be happy to provide any additional information on these contributions will directly emphasize these perspectives in the revision.

"The paper is highly detailed, making it difficult to follow and understand the main contributions that actually lead to improvements in performance and accuracy."

Thank you for raising this concern. We have outlined the main contributions that led to the reported gains in the answer above.

Regarding 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.

For reference, the information regarding the causes of performance increases are currently presented in the following sections:

• 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.
• 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.

We will streamline this in the revision.

Experiments

"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."

We 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 $[1](https://arxiv.org/pdf/2011.01355),[2](https://arxiv.org/pdf/2306.00854)$ .

For 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.

Further, 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.

We would welcome any further discussion on this matter and will clarify the text to emphasize these points.

"Fig6D is disappointing, results produced by the method noticeably differ from the reference"

To 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.

For 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.

Again, 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.

Thank you for engaging in the discussion phase!

We are happy to see that the reviewer has no remaining technical and experimental concerns w.r.t. their original review. Their new concern pertains purely to scope.

“Yet, the contribution is very specific to dMRI. It is probably a better match to a dMRI / Neuroimaging-related conference than to the broader audience in neurips.”

We respectfully disagree and believe that there may be a misunderstanding as,

``

1. NeurIPS has already published several papers on dMRI

dMRI has been explored extensively by both machine learning and neuroscience researchers due to its rich geometric structure and use in measuring neural connectivity.

As a result, NeurIPS has already featured several papers on dMRI analysis with machine learning, demonstrating its relevance to the NeurIPS audience. For example,

• NeurIPS’23
• NeurIPS’20
• NeurIPS’19
• NeurIPS’17
• NeurIPS’14

Additionally, similar conferences and journals in machine learning and computer vision have all featured machine learning work on dMRI data, further supporting its broad relevance to the machine learning community. For example,

• ICLR’23
• CVPR’24
• PAMI’22

``

2. Our work is a general geometric deep learning contribution for spatio-spherical data

NeurIPS/ICML/ICLR and similar venues are strongly interested in geometric deep learning and deep learning on manifolds. However, such manifold structure only arises in specialized applications that may seem niche at first but later become of wide interest to the machine learning community (e.g. geometric deep learning for molecular docking).

While we focus on dMRI in our paper, our work is generically beneficial to the analysis of spatio-spherical signals as it finds several avenues for efficiency gains and builds a framework for sparse non-negative spatio-spherical deconvolution. We foresee several potential benefits in contexts where spatio-spherical data arises: robotics, neural rendering, gaussian splatting, molecular dynamics, etc.

``

3. NeurIPS invites interdisciplinary work in its call for papers

NeurIPS explicitly encourages interdisciplinary submissions in its Call for Papers.

Our work lies at the intersection of the core “Machine learning for sciences (life sciences)” and “Neuroscience and cognitive sciences” areas mentioned in the call as it directly contributes:

• New geometric equivariant deep learning methods for a core life sciences imaging modality (dMRI).
• A novel self-supervised non-negative deconvolution formulation on a spatial graph of spherical signals.
• Enhanced neuronal fiber recovery, which is crucial for the core neuroscience task of understanding brain connectivity.

``

4. Neuroscience is a core topic at NeurIPS and dMRI is the main tool for understanding neural connectivity

Neuroscience has been central to NeurIPS from its inception and understanding the structural connectivity of the brain in vivo relies entirely on dMRI, among having other applications such as surgical planning.

Therefore, in addition to the geometric deep learning community, we foresee our work being of interest to NeurIPS’ neuroscience community as well as our work contributes new deep learning methods that significantly advance the analysis of such data. In turn, these deconvolution advancements lead to more accurate neural pathway estimation that can potentially improve downstream neuroscientific and biomedical analyses.

Thanks again for your engagement!

Thank you for the encouraging feedback! Broadly, we will incorporate all of the detailed suggestions and address the remaining high level questions/comments below:

_”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.”

We 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.

As 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.

We 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.

"I have a small number of suggestions/mistakes enumerated below."

We greatly appreciate the detailed enumeration of typos and awkward phrasings! We will make all suggested changes.

"[...] 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 [...]?"

Thank 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.

However, 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.

As 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.

However, 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.

Thank you for the highly detailed and valuable feedback! We will incorporate the suggestions and address the other concerns below:

Weaknesses

"The main weakness of this manuscript is in the way the contributions section is written at the end of the Introduction. [...]"

Thank you for the detailed suggestions on editing this paragraph! We will incorporate them into the revised paper.

"Some of the contributions (specifically, the in vivo qualitative results) are not present in the main manuscript, but are part of the appendix."

As 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.

However, 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.

Questions

"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 [...]?"

Thank 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.

As 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.

In 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.

However, 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.

We will clarify the results to include this discussion.

"[...] lengthier discussion on how the proposed method would perform on other in vivo clinical datasets, under different noise levels, motion, etc."

We 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.

Briefly, 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.

"[...] slight increase in FPR in both Fig 6A and 6B when comparing SHD(-TV) with RT-ESD."

The reviewer is correct that there is a slight increase in the FPR. Regarding 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. For 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.

"In Fig6C can you explain the “narrowness” of your result as compared to the ground truth or CSD?"

We 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.

Writing improvement suggestions

Thanks again for all of the detailed writing and presentation suggestions. We will incorporate all of them into the revision.

Thank you for the positive evaluation and for highlighting the importance and quality of our methodology, presentation, and experiments.

Antipodal symmetry assumption

"It would be better to improve the flexibility of the model so that it could be applied to diverse scenarios."

We 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.

We 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.

"Can the authors clarify the limitations of the antipodal symmetry assumption?"

Thank 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.

The 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.

More diverse clinical conditions

"Diverse clinical conditions can be considered in future studies, involving varying levels of image quality and pathological changes [...]."

Thank 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.

In 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:

• fODF estimation analyses with diverse image qualities, quantitatively on the DiSCo and Tractometer datasets, as well as qualitatively on the HCP dataset.
• Downstream tractography task analyses on the Tractometer dataset.
• Speed and memory analyses.

Regarding the specific scenarios mentioned by the reviewer,

Brains with lesions

In 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.

However, 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.

Varying image qualities

Regarding 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:

• 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.
• The self-supervised training framework allows the model to be fine-tuned for any new image and imaging setup.

We 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.