Topology-Aware Learning of Tubular Manifolds via SE(3)-Equivariant Network on Ball B-Spline Curve

Sincerely appreciate your thorough review and providing us with invaluable, constructive comments to greatly improve the quality of our manuscript. The response to Weaknesses is as follows:

W1: Limited Novelty

Q1: The key theoretical innovation of this work lies in the use of a compact and continuous representation based on the BBSC manifold, instead of the commonly used discrete forms. Our approach enables us to leverage SE(3)-equivariant mappings to preserve feature consistency under rigid transformations while encoding anatomical constraints through graph topology. The synergy between geometric equivariance and anatomical priors enhances the model’s robustness to variations in orientation, position, and topology, without sacrificing geometric accuracy.

We emphasize that our work presents a new, effective, and efficient paradigm for modeling tubular structures. We outline the key novelties as follows:

1. We employ the compact representation of BBSC manifold to model tubular structures. While BBSCs have been previously studied, their application to real-world tasks has been limited. Compared to voxel or point cloud representations, BBSCs significantly reduce data complexity, lowering storage and computational cost, accelerating convergence, and improving model stability. Moreover, we develop a rigorous mathematical framework for BBSCs by defining a BBSC manifold with a Riemannian metric (see Section 4.1) to measure distances between shapes. We also introduce connection structures to represent complex tubular topology via anatomical connectivity.

2. Modeling in a continuous parameter space offers novelty and practicality. Although SE(3)-equivariant networks have been explored in geometric deep learning, applying it in the continuous BBSC parameter space is a novel contribution. Unlike models operating on discrete data, our approach enables the network to learn precise geometric features, improving generalization and reducing overfitting.

3. We use the strategy to analyze complex vascular topologies. We first segment individual branches and extract their anatomical connectivity. We then study the geometry of each branch and the topology of their interconnections separately, rather than encoding both into the same graph structure. This simplifies the graph while retaining global topological relationships and allows more effective learning of branch-level geometry. Compared to prior studies that represent branches as edges and bifurcation points as nodes, our design is simpler and more expressive.

W2: Narrow Scope and Insufficient Evaluation

A2: Thank you very much for your valuable suggestions regarding the application scope and evaluation of failure cases. We would like to offer the following clarifications:

The clinical MRA data used in our study contain noise, artifacts, and minor peripheral branches that may affect curve fitting. Our preprocessing includes denoising [1], segmentation [2], structure repair [3, 4], and pruning of irrelevant branches. As our focus is on BBSC-based representation, SE(3)-equivariant modeling, and GCN-based topological learning, we did not elaborate on preprocessing [5]. Per your suggestion, we are preparing a more detailed description to improve clarity. Regarding the scope of validation, our work focuses on the CoW, a clinically significant but underexplored structure, to demonstrate our method’s effectiveness. Nonetheless, we agree that broader evaluation is important and are working to extend our method to other tubular anatomies. We also appreciate your comment on failure cases. We have included misclassification examples in the supplementary material, where topological variations lead to errors despite accurate BBSC fitting. We acknowledge the lack of BBSC fitting failures and will add such cases and analysis in the revised manuscript.

W3: Lack of Medical Relevance Testing

Thank you so much for the invaluable insights. Our response are as follows:

1. The CoW branch classification task is clinically important due to its complex topological heterogeneity, but research is limited by the lack of large-scale public data and the challenge of the task itself. Few models exist for this task, and most methods are not open-source. These factors underscores the need for a robust CoW branch classifier. While individual CoW structures vary, the shared geometric and topological features of tubular anatomy make our model transferable to other medical applications, which we are actively exploring.

2. Despite anatomical differences, many tubular structures in medical imaging share common geometric and topological traits. This generalizability allows our framework to be adapted to various anatomical structures beyond CoW. We intend to validate our model in broader medical contexts.

3. CoW classification is challenging due to high inter-subject topological variability from genetic and anatomical differences, along with limited high-quality public datasets. Although existing methods perform well on small benchmark datasets (see Table 1), our results on a large, real-world clinical dataset (see Table 2 and Figure 5) reveal substantial performance gaps and strong overfitting in many baselines. In contrast, our method remains stable and generalizes well, underscoring that CoW classification is a non-trivial task and further validating the strength of our approach.

We respond to your questions as follows:

Q1: How is the BBSC fitting performed?

A1: We fit BBSC from surface point clouds and centerlines via an optimization-based approach. Before fitting, we denoise the original MRA voxel data, allowing our pipeline to robustly handle noisy tubular geometries. For complex multi-branch structures, we decompose them into simpler tubular segments while recording their topological connections, and then fit each segment with an individual BBSC, as demonstrated in our modeling of the full CoW ring structure. Additional relevant reference is [6].

Q2: No real-world biomedical datasets included.

A2: We’d like to clarify that our method has indeed been validated on real-world clinical data. Specifically, Table 2, Figures 4 and 5 all present results based on CoW data collected from real patients. As shown in these experiments, the classification task on real-world data is significantly more challenging than the public TopCoW dataset, due to greater anatomical variability and the presence of more complex topological patterns in clinical cases.

Q3: Can the method be adapted to cases with bifurcations or looped topologies?

A3: Thank you for this insightful question. Our method is indeed capable of handling bifurcations and looped topologies. In fact, CoW itself inherently exhibits both bifurcated and looped vascular structures. To address bifurcations, we adopt a segment-wise BBSC fitting strategy, where complex branches are decomposed into simpler tubular segments while preserving their topological connections. Looped structures naturally correspond to cycles in the constructed graph. Our graph-based representation and the use of GCN can effectively model and learn from such topological features. This design also ensures the scalability of our approach to a wider range of anatomical structures beyond CoW.

Q4: No camparison with MLP and GCN.

A4: This is an excellent question. Since B-spline control points and knot vectors are ordered sequences without complex connectivity, sequence models are more suitable than graph models. In our ablation study (Table 1 and 2), we tested a Transformer, which better models sequences than MLP, and evaluated its impact. We would be happy to include further comparisons upon request.

Thank you for these invaluable constructive suggestions. We will respond to your listed limitations and incorporate the limitations you pointed out into our revised manuscript.

L1：Lack of end-to-end learning.

A1: This is an inspiring suggestion. As we focus on the classification task comprising CoW, so that we did not include the preprocessing part. We intend to talk about the end-to-end learning in the revised manuscript.

L2: Dependence on clean, tube-like structure assumptions.

A2：In fact, the original data we used is noisy; we performed denoising during the preprocessing stage. This work proposes a model architecture for complex tubular structures like blood vessels and nerves. Following your suggestion, we plan to extend our framework to parameter spaces and SE(3)-equivariant networks for non-tubular structures.

L3: No indication of computational cost or efficiency.

A3: We evaluate the inference time and model parameters in Table 3. Our compact parameterization speeds up convergence and reduces storage and computation, making our method fast and lightweight during both training and testing.

L4：Lack of applicability to non-tubular data.

A4: This work aims to propose a model architecture for handling complex tubular structures such as blood vessels and nerves. We also plan to extend our framework to parameter spaces and SE(3)-equivariant networks for non-tubular structures.

L5: No empirical evidence or discussion of the ethical implications of automated anatomical modeling.

A5：CoW data share similar geometry and topology with other tubular structures, because we don’t use CoW-specific features, our method is transferable. We will further validate this.

[1].Liu Z, et al.MRI Joint Super-Resolution and Denoising based on Conditional Stochastic Normalizing Flow.

[2].Lv Z, et al.A Parallel Cerebrovascular Segmentation Algorithm Based on Focused Multi-Gaussians Model and Heterogeneous Markov Random Field.

[3].Liu X, et al. Extending Ball B-spline by B-spline.

[4].Zhao Y, et al. G2 Blending Ball B-Spline Curve by B-Spline.

[5].Wang X, et al. Skeleton-based cerebrovascular quantitative analysis.

[6]. Wu Z, et al. Fitting Scattered Data Points with Ball B-Spline Curves using Particle Swarm Optimization.

I thank the authors for a detailed answer.

I acknowledge the novelty in using BBSC manifold to model tubular structures but I don't think there is novelty beyond this. Having said that, I find this contribution reasonable and interesting.

While I understand the difficulty in finding proper testing data, it is still a significant concern to me.

Everything considered, my overall rating to the paper slightly improved, but not to the point I can confidently champion the paper. This is also partly because I am not familiar with the topic.

Thank you very much for your response and for recognizing our work. Below, we provide some explanations regarding your comments:

First, in terms of innovation, one of our key contributions lies in viewing BBSC from the perspective of manifolds, which you have acknowledged as well. Furthermore, we are the first to implement SE(3)-equivariant mappings on a BBSC manifold parameterized in this way. From the perspective of functional analysis, this can be understood as performing SE(3)-equivariant transformations on a function space, which is fundamentally different from most existing works that implement SE(3) equivariance on 2D images, 3D point clouds. Because the control points of BBSC are ordered coordinates distributed in $\mathbb{R}^3$, forming sequences of 3D points, the existing SE(3)-equivariant networks, which mostly graph-based(e.g., TFN, SE(3)-Transformer, EGNN), cannot be directly applied. To address this, we designed a method that not only preserves SE(3) equivariance but also enables message passing and aggregation among the ordered control points.Additionally, by leveraging the properties of BBSC and their relationship with B-spline parameters, we proposed a strategy that integrates the knot vector and radius. This strategy is applicable to all models based on B-splines and provides a valuable reference for the future design of B-spline SE(3)-equivariant networks, which we consider another major innovation of our work.

Second, Regarding your concern about the test data, we conducted experiments on both public datasets and more complex and practically relevant clinical datasets. For your reference, we can also provide the URL for the TopCoW dataset (https://topcow24.grand-challenge.org/data/). For all types of tubular medical data (e.g., blood vessels), regardless of the presence of noise or complex structures, our pipeline which is introduced in our Weeknes2 response can process and fit BBSC effectively. We believe that validation on clinical datasets is more challenging and more indicative of the practical value of our model compared to public datasets. Therefore, we validated our pipeline using real-world clinical cases. In fact, compared to TopCoW, clinical case data exhibit significantly different topological and geometric structures, accompanied by more complex heterogeneity as well as inevitable imaging noise and detail loss during scanning. This complexity led to severe overfitting in the baselines we compared against, whereas our pipeline was still able to produce smooth splines (see Supplementary Material) and achieve high-accuracy classification. This demonstrates the strong robustness of our pipeline to tubular data with varying structures and distributions.

Lastly, we would be happy to clarify our topic: our goal is to construct a parameterized, continuously differentiable manifold space based on BBSC, implement SE(3) equivariance on this space to learn accurate geometric features, and use GCNs to analyze the topological structure of the manifold space. We validated our approach on a challenging and clinically significant CoW classification task which currently lacks effective computational methods, thereby addressing an important and complex medical problem. If you have any further questions or concerns, we would be glad to provide additional explanations, and we hope this helps you gain a deeper understanding of our work and acknowledge the effort we have invested. We sincerely appreciate your review and valuable feedback, and we look forward to your response.

Thank you so much for your thorough review and providing us with invaluable, constructive comments to greatly improve the quality of our manuscript.

In addition, thank you for pointing out the spelling and formatting errors in the manuscript (mainly due to hasting submission before deadline). We have now carefully proofread and corrected all such issues, including but not limited to equation references and appendix citations. Please find below our responses to all your comments.

Regarding the mathematical formulation:

Q1: Smoothness of mixed partial derivatives.

A1: Per the definition of B-spline curves abbreviated as $\beta$ (Equation 1), the knot vector $\tau$ only appears in the basis functions $N_{i,p}(t)$, while the control points $P$ are multiplied directly by these basis and subsequently summed. If we focus on a specific coordinate component of the control point $P_{i}$, the resulting function $\beta$ is essentially a product of $P_i$ and a function that depends solely on $\tau$. Therefore, $\beta$ can be viewed as a separable multivariate function made up by $P_i$ and $\tau$, and its mixed partial derivatives can be written as the product of the respective partial derivatives. In the appendix, we prove that $\beta$ is infinitely differentiable with respect to both $\tau$ and $P_i$, which implies that all their mixed partial derivatives are continuous. This establishes the smoothness of the BBSC formulation with respect to both $\tau$ and $P_i$. Furthermore, the mixed partial derivatives of the sum of these products will also be continuous, due to the linearity and continuity of differentiation. Similarly, we can show that the mixed partial derivatives of any order with respect to $\tau$ and the control radius $r$ are also continuous. Moreover, $P$ and $r$ do not co-occur in the same term in the BBSC expression. In fact, the pair $(P, r)$ can be interpreted as a 4D B-spline control parameter, where the radius $r$ (the fourth component) is linearly independent of the spatial coordinates represented by the first three components in $P$.

Q2:  Non-standard usage.

A2: We would like to clarify that $\beta$ is infinitely differentiable with respect to $r$, and all of its derivatives are continuous.

Q3: Smoothness of BBSC.

A3: BBSC is locally homeomorphic to $\mathbb{R}^2$, and we denote it as $\beta$ in our manuscript. It is constructed using B-spline basis functions $N_i(t)$ and control parameters $C_i = (x_{i_1}, x_{i_2}, x_{i_3}, r_i) \in \mathbb{R}^4$, where $(x_{i_1}, x_{i_2}, x_{i_3})$ and $r_i$ represent spatial position and radius. In $\mathbb{R}^3$, $\beta = \sum N_i(t) C_i$ defines a tubular surface (see Figure 1). According to B-spline theory, $\beta$ is smooth and all derivatives are continuous. We adopt a piecewise fitting strategy to ensure each segment is regular—non-self-intersecting, free of abrupt directional changes, and non-degenerate. This ensures that $\beta'$ is continuous and non-zero. By the inverse function theorem, $\beta$ is locally injective with a continuous inverse, and thus locally homeomorphic to $\mathbb{R}^2$. $\beta$ is also a smooth function of $(P, r)$ and $\tau$, with constraints on $\tau$ imposed during fitting to maintain regularity. A template BBSC is also used to guide the parameters, ensuring local homeomorphism.

Q4: Bijectivity. 

A4: Thank you for sharing your confusion about our description on “$\beta$ is a linear combination of $P$ and $r$”. Indeed, we could do a much better job to provide comprehensive and precise context. That expression was intended to articulate that in Equation 1, both $P$ and $r$ are combined through a weighted sum with the B-spline basis functions as coefficients. As we clarified in the previous response, $\beta$ is a shorthand notation for the BBSC manifold mentioned in Definition 1. Because BBSC can be regarded as a B-spline with 4-dimensional control parameters, we refer to the theoretical framework presented by De Boor in “A Practical Guide to Splines”. When the spline degree is fixed and the knot vector $\tau$ is constrained to be clamped at both ends and strictly increasing in the interior, the basis functions $N(t)$ are determined recursively through a fractional structure based on $\tau$, which guarantees a unique B-spline curve.

Q5: How would the GCN preserve equivariant information?

A5: We sincerely appreciate your constructive and insightful suggestions. We believe SE(3)-equivariance captures geometric symmetries of individual branches, while graph convolution handles topological invariance between branches—two aspects that should be treated separately. That said, integrating SE(3)-equivariance into GCN can improve model robustness. To reduce equivariance loss, we design the graph aggregation to preserve SE(3)-equivariance as much as possible. Specifically, we use only concatenation and relative differences between neighbors, which preserve SE(3)-equivariance, besides the MLP. Though equivariance loss incurred by MLP, our results show that the architecture remains effective.

Regarding your concerns about the experimental results:

Q1: Confusion of the CoW size.

A1: We apologize for any confusion regarding the dataset description. The TopCoW dataset contains 125 cases, each with vessel segmentations and branch-level labels. Our method learns both the geometric features of individual branches and the topological relationships among branches within the same vessel. Figures 3 and 6 show accuracy and label distributions. As stated, we perform classification using the provided ground-truth labels to validate our approach.

Q2: Why was cross-validation not performed on the clinical dataset?

A2: Because the TopCoW 2024 dataset contains only 125 CoW cases, which is relatively small in scale, we performed 5-fold cross-validation to ensure the reliability of our results and to mitigate overfitting. In contrast, the size of our clinical dataset is nearly 9 times bigger. Thus, it provides sufficient data to train the models effectively and reduce the risk of overfitting. Therefore, we did not adopt cross-validation for this dataset. We are happy to include cross-validation experiments on the clinical data. We have updated cross-validation of our method on the clinical data, and the results are as follows:

Accuracy：$96.11 \pm 1.01$; AUC-ROC: $99.38\pm0.19$; Precision: $96.11\pm0.93$; Recall: $96.08\pm0.97$; F1 score: $96.02\pm1.05$

Q3: Description of the CoW dataset is unclear.

A3: For the clinical dataset, we directly deal with the original MRA voxels image. The Diffusion model is used to denoising and regularize the whole dataset [1]. The Fast RCNN [2] is used to detect the CoW part of the overall model. The statistic model is used to segment the brain vessel with Markov constraints [3]. Then BBSC is fitted by the strategy introduced in Section 4.2. To deal with the broken part of the branch, we use the extension [4] and blending methods [5] on the BBSC for repair. The overall pipeline together with the geometric morphological quantification can be found in [6,7]. The graph data was constructed by extracting the connectivity relationships between the segmented branches. We are revising the manuscript to provide a more thorough, clearer explanation about these preprocessing steps.

Q4: The number of clinical data between the main text and the appendix is inconsistent.

A4: We did collate 1,128 real-world clinical scans. However, these raw data had not been quality-checked in advance. A few scans exhibited severe artifacts such as motion blur, image corruption, or duplication, making them unsuitable for normal processing. 28 cases were excluded following the suggestion of the radiologist. This quality check does not affect the reliability of our experiments or the randomness of our sampling. Anatomical variations in real-world clinical data, such as missing CoW branches, cause class imbalance. In TopCoW, branch label 12 appears in only 6 cases. Random sampling could place all such cases in training, leaving none for testing. To avoid this, we employed stratified sampling based on branch label distribution to ensure that each test set contains at least one case with label 12. For the clinical dataset, the large dataset mitigates this issue. Hence, we adopted a standard random sampling and splitting.

Q5: Baseline methods in related work are not compared.

A5: In comparisons, we included methods from the related work section that can be applied to the CoW classification task, such as SE(3)-Transformer (a variant of TFN) and PointNet. While many other methods reviewed in the same section are relevant to SE(3)-equivariance, they cannot be directly applied to our task. For example, DSCNet is designed for 2D image data, DDT is a segmentation model, and GeoTransformer focuses onregistration using distance and angle as inputs.

Q6: Regarding NeurIPS checklist:

A6: For Q7, Table 1 reports the error bars and Figure 6 in Appendix D.1 visualizes the statistical significance of the experiments.

The following are the references (due to space constraints, only the titles are provided).

[1].Liu Z, et al.MRI Joint Super-Resolution and Denoising based on Conditional Stochastic Normalizing Flow.

[2]. Girshick R. Fast r-cnn.

[3].Lv Z, et al.A Parallel Cerebrovascular Segmentation Algorithm Based on Focused Multi-Gaussians Model and Heterogeneous Markov Random Field.

[4].Liu X, et al. Extending Ball B-spline by B-spline.

[5]. Zhao Y, et al. G2 Blending Ball B-Spline Curve by B-Spline.

[6].Wang X, et al. Skeleton-based cerebrovascular quantitative analysis.

[7].Y Liu, et al. Automated anatomical labeling of a topologically variant abdominal arterial system via probabilistic hypergraph matching.

The reviewer would like to thank the authors for their extensive response and also for their extensive responses to the other reviewers. Appreciation is also extended for their additional proofreading and updating of the manuscript. While the answers rectify the confusion for the reviewer, the fact that the reviewer is unable to view this updated version of the manuscript, puts them in a difficult position. With revisions of such profound nature the reviewer will not be absolutely certain that the paper is clear, finished and in a publishable state. Therefore the reviewer has come to the unfortunately conclusion that raising their score at this stage will be difficult.

That said, they will lower their confidence. In case the other reviewers and AC are convinced this work should be published, that allows this course of action to be taken. The confidence of the reviewer will be lowered to a 3 instead of a 4, unfortunately this can not be modified through OpenReview.

The reviewer hopes for the understanding of the authors.

Thank you very much for your thorough review and providing us with many invaluable suggestions. We also appreciate the opportunity to clarify the concerns you raised. Please find below our clarifications and explanations regarding the weaknesses you raised.

W1: Not much topological discussion A1: We sincerely thank you for your suggestions regarding our topology-aware design. Our approach parameterizes each vessel branch as a BBSC, treating it as a differentiable manifold and modeling it as a graph node. The edges in the graph are defined by the connectivity between branches. This allows us to avoid the complexity of edge attribute modeling and instead focus on learning the rich geometric features encoded in the BBSCs as nodes. We believe this is a novel and effective strategy for certain tasks. Furthermore, as described in Definition 2 in Section 4.1, we define a Riemannian metric between BBSCs to facilitate comparisons between different branch representations.

In contrast to most existing models that analyze the topological structure of vascular by constructing graph edges from vessel branches and defining graph nodes as junction points between branches, e.g., ”Statistical Analysis of Complex Shape Graphs” segments complex tubular structures into branches represented as edges and uses branch connection points as nodes, while employing metrics like effective resistance, elastic shape analysis, and geodesic distance in elastic shape space to compare different graph shapes. Our method has the advantage of preserving the original topological connectivity while offering stronger learning capability and interpretability of the geometric features within each branch.

W2: The significance of SE(3)-equivariant to performance is unclear. A2: Thank you for pointing out that the results in Table 1 suggest SE(3)-equivariance may not significantly improve performance. This is a very reasonable observation. However, we would like to clarify that the TopCoW task in Table 1 is based on the public dataset provided by the 2024 MICCAI Challenge, which contains 125 high-quality but limited CoW branch samples with 12 annotated categories. Compared to the task in Table 2, the TopCoW dataset is relatively simple. The high performance are achieved by most baselines, which makes it difficult to highlight the benefits of SE(3)-equivariance. When evaluated on a more challenging and larger-scale clinical dataset, the advantages of SE(3)-equivariance become more apparent. Specifically, the number of categories increases to 22 and the data volume grows (approximately 9 times larger), SE(3)-equivariance leads to noticeable improvements across multiple metrics (with the only exception of AUC-ROC). Furthermore, we discuss how SE(3)-equivariant models exhibit greater resistance to overfitting compared to other baselines (see Figure 5). Through comparison, we observe that on more complex real-world clinical data with intricate bifurcations, incorporating SE(3)-equivariance significantly enhances model stability and performance. This further demonstrates the effectiveness of SE(3)-equivariance in handling complex geometric features.

W3: Is the input data relatively clean? A3: Thank you very much for your invaluable comment. We fully agree with you on this. In fact, our original input data consists of raw MRA scans, which often contain noise and complex branching structures that can negatively impact the fitting of BBSCs. During preprocessing, we first applied denoising techniques to the MRA volumes[1], then extracted surface point clouds for further refinement. For data where vessel branches are not pre-segmented, we first apply a segmentation model (e.g., [2]) to extract the branches. Then, we construct BBSCs using the method described in this submission. During the fitting process, BBSC extension [3] and blending [4] techniques may also be applied when necessary. However, our primary goal in this work is to highlight the modeling of tubular structures via BBSC, along with the learning of geometric features through SE(3)-equivariant networks and topological relations through graph neural networks. Therefore, we did not elaborate on the data preprocessing pipeline in detail. We appreciate your suggestion and are preparing a more comprehensive description of the preprocessing steps.

We sincerely appreciate your interest in our work. We address your questions as follows.

Q1：Can you provide some examples of the input data?

A1: We utilized the TopCoW 2024 dataset, publicly available at https://topcow24.grand-challenge.org/data/, which was introduced by the challenge organizers in the paper Benchmarking the CoW with the TopCoW Challenge for CTA and MRA [5]. Specifically, we used the pre-segmented and branch-labeled subset of the data for model training and evaluation. We would like to share the parameters for one example branch (due to space limitations) in the supply material. The following are the control parameters (x, y, z, radius):

-0.9574 0.5936 0.6443 0.9277

-0.8946 0.5826 0.6481 0.9218

-0.7000 0.5475 0.6357 0.8960

-0.3348 0.1458 0.5663 0.9499

-0.2276 -0.0279 0.5510 1.0042

0.0665 -0.3906 0.4303 0.9124

0.3661 -0.5725 0.1682 0.9839

0.5042 -0.5007 -0.0663 0.9944

0.6324 -0.2403 -0.3195 1.0272

0.6850 -0.0188 -0.4251 0.9941

0.7426 0.2168 -0.5108 1.2511

0.7992 0.3759 -0.5876 2.1610

0.8091 0.4154 -0.6635 0.8286

The corresponding knot vector is: [0.0, 0.0, 0.0, 0.0, 0.0357, 0.0646, 0.1030, 0.1692, 0.2370, 0.2833, 0.3230, 0.3530, 0.4215, 1.0, 1.0, 1.0, 1.0]

Q2: Comparison with shape graph representations.

A2: We agree that shape graph representations is a very interesting and valuable direction to explore. We appreciate your suggestion to include comparisons with such methods. We plan to extend our experiments to include quantitative and qualitative comparisons with representative shape graph-based approaches to better situate our method within the broader context in the revision.

Q3: Discussion around the cost function and data matching term

A3: Thank you for this excellent question. The sum of the Riemannian distance defined in Section 4.1 and the bidirectional Hausdorff distance serves as the cost function, where the bidirectional Hausdorff distance specifically acts as the data matching term you mentioned. The details are as follows:

During BBSC fitting, we first extract the vessel surface point cloud and estimate the centerline from the MRA voxel data. A template BBSC is initialized by fitting a spline to the centerline and setting a constant radius based on the average distance from the surface point cloud to the centerline. This template provides a prior shape constraint (e.g., no self-intersection) using the Riemannian metric described in Section 4.1. To evaluate the matching between the fitted BBSC and the input data, we uniformly sample points on the BBSC surface and compute the bidirectional Hausdorff distance between these samples and the original surface point cloud. This acts as the data matching term during BBSC optimization. For the classification task, we use a standard cross-entropy loss. If one is looking for a specific measure of how well the fitted BBSC aligns with the original vessel data, the Hausdorff distance between the BBSC surface samples and the extracted vessel surface point cloud is an effective and practical metric.

[1].Liu Z, et al. MRI Joint Super-Resolution and Denoising based on Conditional Stochastic Normalizing Flow.

[2].Lv Z, et al. A Parallel Cerebrovascular Segmentation Algorithm Based on Focused Multi-Gaussians Model and Heterogeneous Markov Random Field.

[3].Liu X, et al. Extending Ball B-spline by B-spline.

[4].Zhao Y, et al. G2 Blending Ball B-Spline Curve by B-Spline.

[5].Yang K, et al. Benchmarking the cow with the topcow challenge: Topology-aware anatomical segmentation of the circle of willis for CTA and MRA.

We sincerely appreciate your positive comments on our work and the invaluable suggestions provided. We address your insightful questions point by point as follows:

Q 1: Ambiguity in topology and mechanism to handle it.

A1：The issue you raised is indeed very important and frequently encountered when dealing with medical imaging data such as CT and MRA scans that exhibit complex and sometimes incomplete vascular topology. Specifically, missing branches are common anatomical variants in the CoW, and many such cases are present in our dataset. We’d like to describe how we handle such cases, with reference to related work. We use MRA scans as the raw input data for our processing pipeline. The diffusion model can be used to denoise and regularize the voxel dataset with occlusions and noise [1]. Fast R-CNN[2] can effectively extract the CoW from noisy MRA data. A variety of machine learning and deep learning methods have been developed for vascular segmentation, including statistical models based on Markov constraints [3]. In the BBSC fitting process, we first extract a point cloud from the voxel data, followed by further denoising. Point cloud denoising techniques are already well established, with methods such as PointASNL demonstrating strong performance. For broken parts of vessel branches and incomplete skeleton extracting, we apply an extension and blending strategy [4, 5] for repair. In addition to the BBSC fitting method described in the paper, Zhao Y, et al [5] also provides a reference algorithm for BBSC fitting. A representative example of such a pipeline can be found in [7, 8]. Graph representations and GCN can also partially address topological issues such as missing branches and occlusions. Our experimental results demonstrate that the proposed GNN exhibits a certain degree of robustness in handling such variations. Leveraging such techniques allows us to preprocess the data and recover relatively accurate topological structures even in the presence of missing branches, occlusions, or noise in the CoW data.

Q2: Overlook of the SE(3)-equivariant in other areas of medical imaging .

A2: Thank you for highlighting the missing discussion on SE(3)-equivariance in other medical imaging domains. This valuable suggestion helps improve readers’ understanding of its relevance and effectiveness.

Human anatomy shows strong symmetry and self-similarity. Medical images often involve rigid transformations due to patient positioning, organ motion, or device movement. For example, brain structures are symmetric and similar across individuals. These properties make SE(3)-equivariant networks well-suited for modeling and analyzing such data. We have expanded our literature review to highlight practical applications of SE(3)-equivariance. For example, Billot B, et al [9] proposed the construction of equivariant spatial means using steerable CNN filters and introduced an innovative use of self-attention mechanisms to process temporal sequences. This method achieved important breakthroughs in measuring, tracking, and correcting fetal MRI motion. Wang J, et al. [10] developed EquiTrack, a steerable E-CNN with spherical harmonic kernels and a denoising front-end, setting a new benchmark in brain MRI motion correction. SE(3)-equivariance has also found applications in bioinformatics [11]. We plan to continue surveying related work and improve our coverage in future revisions.

Q3: Whether the geometry and topology have learned clinical variables. Can the method be used as a biomarker?

A3: Thank you for raising this important point regarding the clinical applicability of our model. Indeed, your question touches on a critical aspect of real-world deployment. This is very helpful to us. In our current implementation, the only clinical variable used is the vascular branch label, which serves as the supervision signal during training. Our goal is to develop a general framework that focuses primarily on geometric and topological features, potentially applicable beyond the medical imaging domain. However, in the context of CoW branch classification, we argue that the geometric variations and topological heterogeneity among different branches are themselves highly informative clinical features. For example, the branches labeled ICA-L and ICA-R typically exhibit stronger curvature and high bilateral symmetry. Meanwhile, PCA-L and PCA-R branches are topologically connected only to PCom-L, PCom-R, and the basilar artery (BA), which are critical relationships for accurate anatomical segmentation and diagnosis in clinical practice. Furthermore, we have conducted experiments on real-world clinical data in collaboration with a hospital. These studies demonstrate that our method has the potential to serve as a biomarker or as part of a clinical decision-support tool, highlighting its promise in practical applications. In the future, we plan to incorporate clinical knowledge and extend our general framework to real-world clinical applications, leveraging additional biological markers to achieve more accurate segmentation, classification, and analysis.

Q4: Trade-offs between smoothness and accuracy.

A4: Your concern regarding the trade-offs between smoothness and topological accuracy is very reasonable.Sincerely appreciate this important and meaningful question. The answer to this question is that we balance the smoothness and topological accuracy of the learning during the whole methodology.

1. Our use of B-spline-based parametric representation to achieve smooth and differentiable modeling does not compromise topological accuracy. On the contrary, it improves the model's classification accuracy and accelerates both convergence and inference speed(Figure 5 and Table 3).

2. Our method models each vascular branch individually using BBSC, and the topological relationships are learned across branches. Therefore, representing individual branches with BBSC does not interfere with the learning of inter-branch topology. At the same time, BBSC enables more accurate estimation of branch-level geometric features, which is particularly helpful for classifying branches with complex geometry. Moreover, because the branches inherently follow certain topological priors, the accuracy of geometric representation contributes to topological accuracy. In this sense, the precise geometry captured by BBSC actually helps the model learn topological structures more effectively, thus improving overall accuracy.

3. BBSC offers a compact representation using only a small set of control points, control radius, and a corresponding knot vector, replacing dense point cloud representations of tubular structures. Compared to processing large-scale point clouds, operating on this compact set of parameters significantly speeds up training and inference and also improves model stability.

Thank you for your thorough review—your suggested limitations are very helpful and will guide our future improvements; please find our responses and clarifications as follows.

L1: Complexity and training stability

A1: Thank you for pointing out the potential limitations regarding training stability, failure cases, and convergence behavior. In our framework, the BBSC construction process is decoupled from the training of the SE(3)-BBSCFormerGCN model. The lightweight parametric representation of BBSC in fact accelerates model convergence and enhances training stability. Our model demonstrates faster convergence and relatively stable training behavior on clinical datasets, along with strong generalization and resistance to overfitting(see Figure 5). We would like to sufficiently clarified this point in the revised version. To address this, we have included visual examples of representative failure cases in the supplementary material, and we are happy to provide more detailed illustrations and analyses on convergence speed, training stability, and failure scenarios upon request.

L2: Scalability to noisy or incomplete data

A2: Thank you for highlighting the limitation regarding noisy and incomplete data—this is very helpful for improving our model. Indeed, we encountered such cases in MRA data and addressed them to some extent through denoising, segmentation and reconstruction. As this work focuses on proposing a BBSC and SE(3)-equivariant framework for learning local geometry and global topology, we did not elaborate on preprocessing, but we plan to detail the full pipeline in revision.

L3: Limited discussion of biological plausibility A3: Thank you for pointing out the lack of discussion on biological plausibility, which is indeed important for medical tasks. As noted in Limitation 2, our current focus is on proposing and validating a model for learning geometric and topological features of complex tubular structures, and we plan to address biologically and clinically constrained tasks in future work.

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