Title: Topology-Aware Learning of Tubular Manifolds via SE(3)-Equivariant Network on Ball B-Spline Curve
Venue: NeurIPS 2025
Accepted: yes
Keywords: Tubular manifold

SE(3)-equivariant network，Graph neural network

Vessel classification
Abstract: Tubular-like system shape analysis is quite difficult in geometry and topology, while it is widely used in plants and organs analysis in practice.
However, traditional discrete representations such as voxels and point clouds often require substantial storage and may lead to the loss of fine-grained geometric and topological details. 
To address these challenges, we propose SE(3)-BBSCformerGCN, a novel framework for learning shape-aware representations from continuous tubular topological manifolds with equivariance to rotations and translations.
Our approach leverages Ball B-Spline Curve (BBSC) 
to define tubular manifolds and its functional space.
We provide a formal mathematical definition and analysis of the resulting manifolds and the BBSC functional space, and incorporate an equivariant mapping that preserves geometric and topological stability. 
Compared to the point cloud and voxel based representations,
our manifold-based formulation significantly reduces data complexity while preserving geometric attributes
together with topological features. 
We validate our method on the branch classification task for Circle of Willis (CoW) on the TopCoW 2024 dataset and  the clinical dataset.
Our method consistently outperforms voxel and point cloud based baselines in terms of classification performance, generalization ability, convergence speed, and robustness to overfitting.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a framework for learning from 3D tubular manifolds (e.g., neuron traces or airway structures) by modeling them as Ball B-spline curves. The authors argue that this approach is topology-aware and captures important geometric invariants, leading to improved performance on tasks like classification and curve matching. The pipeline consists of: (1) fitting tubular structures with BBS curves, (2) processing their control points using an equivariant tensor field network, and (3) using the output for downstream tasks.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Geometrically principled formulation: It's an interesting approach. The resulting output is intuitive, interpretable, and controllable - could be useful for downstream tasks.\n\n- Well-motivated application domain: Tubular manifolds (e.g., airways, neurons) are an important and underexplored shape category in deep learning, where topology matters.\n\n- Compact representation: Ball B-spline curves offer a natural, low-dimensional parameterization of tube-like shapes, reducing redundancy in point clouds or voxel grids.\n\nWeaknesses:\n\n1. Limited Novelty\n- Ball B-spline representations are not new, e.g., [1,2,3]. They have been used extensively in graphics, medical modeling, and CAD for tubular shape reconstruction. Use of SE(3)-equivariant networks is well-established in geometric deep learning and the application here is relatively straightforward. The core components of the approach — Ball B-spline curve representation and SE(3)-equivariant neural networks — are not novel independently. While combining them for tubular structure learning is mildly interesting, the integration lacks algorithmic depth or architectural innovation. The contribution is largely in applying known ideas in a new (but limited) context.\n\n2. Narrow Scope and Insufficient Evaluation\n- The method is highly tailored to smooth, tree-like tubular data. It is unclear whether it can generalize beyond clean, noise-free tubes with simple topology. Real-world applications, such as blood vessel or neuron morphology analysis, are mentioned but not evaluated.\n\n- There is no discussion of failure cases, e.g., what happens when the tubular assumption breaks (e.g., loops, knots, branching anomalies).\n\n3. Lack of Medical Relevance Testing\n- Tubular manifolds are most impactful in biomedical imaging (e.g., blood vessels, dendritic trees). The paper does not evaluate on public vessel datasets (e.g., VascuSynth, DRIVE, or Aneurisk).\n\n- Claims of medical relevance (e.g., for airways or neurons) are speculative without quantitative validation.\n\n- The performance gains are modest, and it's unclear whether the method offers enough improvement to justify its added complexity. The task appears to be fairly simple where the accuracy is up to 99.9%.\n\n[1] G2 Blending Ball B-Spline Curve by B-Spline\n[2] SplineGen: a generative model for B-spline approximation of unorganized points\n[3] B-spline curve theory: An overview and applications in real life\n\nQUESTIONS:\nHow is the BBS fitting performed? Is it learned or manually initialized? Can it handle noisy or complex geometries?\n\nWhy are no real-world biomedical datasets included, despite clear motivation from that domain?\n\nCan the method be adapted to cases with bifurcations or looped topologies?\n\nHow does the SE(3)-equivariant model compare to a simpler MLP or GCN over control points?\n\nLIMITATIONS:\nThe authors do not adequately discuss limitations (only 1 sentence in section 6). Some key issues:\n\nLack of end-to-end learning (curve fitting is done externally).\n\nDependence on clean, tube-like structure assumptions.\n\nNo indication of computational cost or efficiency.\n\nLack of applicability to non-tubular data (or even general graph-structured data).\n\nBroader impacts in medical contexts are mentioned, but not substantiated with empirical evidence or discussion on ethical implications of automated anatomical modeling.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-21"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nMedical data such as 3D models of arteries often come with challenges for machine learning. The main obstacle is formed by finding good and effective data representations. Often such data is represented by a point cloud. The authors propose to use a B-Spline approximation to define a tubular neighborhood and present a SE(3)-equivariant network to handle such structures. \nThe method is evaluated on a small dataset from the TopCoW challenge and a private dataset.\n\nSTRENGTHS AND WEAKNESSES:\n**Strong points**\n\n- The idea of using a spline approximation for modeling is novel and has not been attempted thus far. \n- In combination with the SE(3)-equivariant neural network this indeed has the potential to improve \nthe state of the art. \n\n**Weak points**\n\nThe reviewer appreciates the authors for submitting and recognizes the effort required to prepare \na submission for a conference. While reviewing the manuscript a significant amount of spelling \nmistakes and instances of inconsistent formatting were found. The volume poses a concern to \nthe reviewer as it interferes with the quality of the submission.\n\nA small subset of the total typos found in the text are listed below.\n- mainfold -> manifold, consturct -> construct, fuctnion -> function, etc.\n- Abstract\n- Line 41 , 95, 111, 188, 190, 197,  219, 232,  236, 242, 251, 253, 257, 263, 266, \n- (Table 2 and 3 (2x) ) (Method -> method)\n- When typesetting equations, it is recommended to use `\\text{\\cdot}` for operator names such as the real part or the tanh function. \n\nSome notable inconsistencies with respect to referencing. As of now this is not in \nline with the NeurIPS paper formatting guidelines.\n- Equations (expressions to be more precise) are ideally referred to as \"Equation 1.2\" \n\t\tas compared to \"Equation (1.2)\" \n- Appendices are ideally referred to as \"We refer the reader to Appendix D.2\" as compared to \"We refer the reader to D.2\"\n\n\nAdditionally there are multiple mathematically inaccurate, confusing and incorrect statements.\n\n- For a function to be smooth, all partial derivatives need to be continuous, including the mixed partial \n\tderivatives. The authors do not mention this as a requirement and neither prove their continuity in the appendix. \n- The expression $\\frac{\\partial^{\\infty}}{\\partial\\tau^{\\infty}}$ is non-standard usage and \n\trequires additional clarification.\n- To prove that a manifold is smooth, one has to show that the manifold is locally diffeomorphic to \n\t$\\mathbb{R}^{n}$ and proving that a parametrization of the manifold is smooth is not sufficient. For example, \n\tshowing that the map $f:\\mathbb{R}\\to S^1, t\\mapsto \\text{sin}(t)$ is a smooth map does not prove that the manifold $S^{1}$ \n\tis a smooth manifold. This follows from the fact that $f$ is not a diffeomorphism. The proof for the \n\tBBSC manifold to be a manifold will need substantial additions to be a valid proof. Moreover, from the definition\n\tit is not clear what the manifold structure of a BBSC is, and an intuitive sketch and / or clarification \n\twould be helpful for the reader. \n- The proof of bijectivity in Section 4.4 is hard to follow. What do the authors mean with \"$\\beta$ is a linear \n\tcombination of $P$ and $r$\". In the reference to Equation 1, no $\\beta$ can be found. How does injectivity follow \n\tfrom the fact that the quantity $\\beta$ is a \"recursive fractional linear combination of $\\tau$, hence it is \n\tinjective\" (Line 207). \n- \"Finally, the GCN designed to handle topological relationships preserves the overall topological stability of the manifold space,\n\tthereby enhancing the model’s generalization ability and robustness with a lightweight structure\". \n\tFrom the definition of the GCN proposed by the authors, the GCN seems not equivariant with respect to SE(3). A similar \n\tremark is  also made by the authors in the conclusion. How would the GCN be able to preserve any geometrical information \n        learned in the the previous layers if it is neither equivariant nor invariant?\n\n\nThe experimental section has caused confusion with the reviewer. \n- The dataset size for CoW is stated to be \"125 over 12 classes\". The authors \n\tmeant to write that the CoW dataset has on average (roughly) 125  elements per class. \n\tThis can only be deduced by looking at the $y$-axis of Figure 6 in Appendix D.1 and such \nphrasing leads to confusion with the reader. \n\tWhen researching the CoW dataset the reviewer found that the original task was a \n\tsegmentation challenge and not a classification challenge. It would be good to elaborate on \n\thow this segmentation task can be turned into a classification task. \n- What led the authors in the decision to omit a 5-fold cross validation with the clinical dataset? Providing \n\tsuch a setup would significantly increase the quality of the work. If the authors choose to use \n\t5-fold cross validation for one experiment, they should motivate why only reporting a single evaluation \n\tin the other is a reasonable thing to do.\n- The description of the CoW dataset remains unclear to the reviewer. Is it a point cloud or a graph dataset and \n\twhat are the features in the dataset? A detailed analysis can not be found in either the main text nor in the appendix, \n\tleading to difficulties if one would like to reproduce the experiments based on the text and appendices.\n- In the appendix the clinical dataset is split into 900 train samples and 200 test samples. \n\tThe main text states that the clinical dataset consists of 1128 samples. In the figures (see the first point) it seems \n\tthat these are averages per (unbalanced) class. Please note that these numbers do not correspond. Assuming that the \n\tauthors mean on average and per class, how are the samples picked given the imbalance? They mention \n\tstratified sampling, hence additional clarification is needed at a minimum. \n\nProviding the code used in the experiments would also greatly strengthen the text, in particular since the \nmain text is confusing to the reviewer.\n\nIn the related work section the authors name various comparable methods, but none of them is \nused as a comparison partner. Conversely, the comparison partners used in the experimental \nsection are not mentioned in the related work section. What is the motivation behind this \ndecision? If there is related work it is certainly good to include it in the form of comparison partners.\n\nThe reviewer also believes that there are discrepancies between the paper and the answers provided in the NeurIPS checklist. \nMost notably in Q7 the authors state that they report error bars for the experiments supporting the main claims, while this is in \nfact only done for one experiment. The other experiment is an experiment supporting the main claims of the efficacy of their \nmethod. Therefore they should have mentioned it in the checklist. This omission has caused serious concern with the reviewer.\n\nBased on the above the reviewer believes that a substantial amount of improvements \nare required to meet the quality standards of NeurIPS. Moreover an additional round of reviewing would be needed \nto ensure that the proofs are accurate. Since the required improvements are beyond the scope of the \nrebuttal period, the recommendation of the reviewer is to (potentially strongly) reject the paper.\n\nQUESTIONS:\nSee the above section.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 2, "CLARITY": 1, "IMPACT": 1, "ORIGINALITY": 2, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel method for estimating tubular branching structures—modeled as graphs with edges represented by smooth tubular curves—from skeletonized and point cloud data. The motivation stems from anatomical applications such as the segmentation of complex vascular structures. The core idea is to represent these branching graphs using Ball B-spline curves, where both the centerlines and associated radius functions are expressed in spline bases. To estimate the spline coefficients from data, the authors design a pipeline that combines transformer networks and graph convolutional networks, enabling structural and geometric learning. This representation is efficient and geometrically natural compared to traditional voxel- or point cloud-based approaches. Also, the framework incorporates SE(3)-equivariant mappings to ensure invariance to rigid body motions. The method is evaluated on the TopCoW Challenge dataset, where it demonstrates improved performance over prior techniques in terms of classification accuracy and generalization. The experimental results support the utility of the proposed shape-aware and geometry-preserving representation.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\nThe use of graph structure is natural for representing such branching anatomical objects. It provides an efficient representation for seemingly complex objects and paves way to improved estimation (or segmentation) performance. Representing tubular objects (curves and radius functions) by splines or any other basis are natural. With this representation and the pre-processed data (that includes skeletal curves plus point clouds), the actual shape estimation amounts to fitting (curve fitting or tube fitting). The paper provides significant gains on a standard dataset involving segments of anatomical structures. \n\nWeaknesses: \n1.\tAlthough the paper has topology-aware in the title, there is not much topological discussion in the paper. It seems to me that the topological perspective is hidden in choosing the graph-based representation. Similar graph representations for vasculature structures have been used in the past (see e.g. Statistical Analysis of Complex Shape Graphs, https://ieeexplore.ieee.org/document/10552433) although the authors have not mentioned them. The presence of such literature reduces the novelty of the current paper. \n2.\tFrom the ablation study (Table 1, bottom rows), it is not clear if the SE3 equivariance is contributing significantly to the performance. Most of the performance gain seems to come from the spline-based representation and estimation using GCN.  Although SE(3)  equivariance is relevant conceptually, that discussion adds a lot of technical material to the paper without much evidence in the results. \n3.\tThe case study presented in the Supplementary looks interesting, but it does not show what the input data look like. If the input data is relatively clean, then the results will not be that impressive, and vice-versa.\n\nQUESTIONS:\nCan you provide some examples of the input data to your network? \n\nCan you compare your spline representations to the shape graph representations used in previous papers (See a reference above). \n\nCan you clarify (simplify) the discussion around the cost function being used to train the network. I am looking for a data matching term that evaluates the estimated structure against the input data.\n\nLIMITATIONS:\nThe paper mentions the lack of equivariance of GCN but does not discuss other limitations of the pipeline.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper makes a significant theoretical and practical contribution to the field of 3D anatomical modeling, especially for tree-structured objects. It combines geometric deep learning, differential geometry, and anatomical prior knowledge in a compelling and well-structured way. The topology-aware and SE(3)-equivariant framework is novel and well-justified both mathematically and experimentally. The approach leverages a manifold-aware representation that captures the topology and geometry of branching tubular objects in a way that is both rotation-translation equivariant and differentiable. Applications include modeling airways, blood vessels, and other tree-like anatomical structures from 3D volumes or point clouds.\n\nSTRENGTHS AND WEAKNESSES:\n1. Novel Representation:\nRepresenting tubular objects as ball B-spline curves that evolve over SE(3) is a highly expressive and geometrically faithful approach. This avoids discretization errors common in voxel-based methods and preserves topology explicitly.\n2. SE(3)-Equivariance:\nIncorporating equivariance to rigid body transformations ensures that the learned representations generalize well under spatial transformations, which is important for anatomical data with arbitrary orientation.\n3. Topology-Aware Design:\nThe explicit use of a tree-structured latent representation with continuous branching points is a key contribution. It allows for more accurate modeling of bifurcations and curved branches.\n4. Theoretical Rigor and Differentiability:\nThe paper shows that their curve and radius representation is differentiable, enabling end-to-end training. This is important for fitting continuous surfaces and performing downstream analysis.\n5. Strong Empirical Results:\nDemonstrated improvements over baselines such as TreeGCN and Voxel2Mesh in airway segmentation tasks show the method’s practical value. Qualitative reconstructions are anatomically plausible.\n\nQUESTIONS:\n1. How does the method handle ambiguity in topology when the input volume has missing branches, occlusions, or noise?\nIs there any uncertainty quantification or mechanism to handle underdetermined branching structures?\n\n2. How does the use of SE(3)-equivariance in this work relate to its applications in other areas of medical imaging, where it has shown clear practical benefits?\n\nThe paper overlooks a relevant branch of literature where equivariant networks have been successfully applied to handle challenges like motion and misalignment. For example, SE(3)-Equivariant and Noise-Invariant 3D Rigid Motion Tracking in Brain MRI and SpaER: Learning Spatio-temporal Equivariant Representations for Fetal Brain Motion Tracking both demonstrate the value of incorporating SE(3) symmetry in settings such as fetal brain imaging. I suggest to refine the introduction and add discussion of these works would help ground the current method in existing medical use cases and strengthen the motivation for using equivariant architectures.\n\n3. Does the learned latent topology or geometry correlate with clinical variables, such as airway obstruction severity, anatomical asymmetry, or disease subtype?\nCould this method be used as a biomarker or as part of a clinical decision-support tool?\n\n4. The method claims end-to-end differentiability through B-spline parameterizations. Are there observed trade-offs between smoothness and topological accuracy during training?\nDoes enforcing differentiability at the cost of interpretability or discretization flexibility limit downstream tasks?\n\nLIMITATIONS:\n1. Complexity and Training Stability:\nThe method introduces several architectural components (e.g., equivariant encoder, recursive curve decoder, B-spline differentiable renderer) which may require careful tuning. Details about training stability, failure cases, or convergence behavior are limited.\n\n2. Scalability to Noisy or Incomplete Data:\nWhile the model performs well on pre-processed medical volumes, its robustness to noisy segmentations or low-resolution inputs is unclear. How well does it generalize when input branches are fragmented or missing?\n\n3. Limited Discussion of Biological Plausibility:\nThe method models geometry and topology, but does not integrate any biological priors or clinical constraints, such as branch angles, diameters, or plausible tree topology (e.g., for arteries vs. airways).", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-02"}