Topology-Aware Learning of Tubular Manifolds via SE(3)-Equivariant Network on Ball B-Spline Curve
Authors: Jingxuan Wang
Zhongke Wu
Wang Xing-Ce
Zhang Zeyao
Chunhao Zheng
Di Wang
Venue: NeurIPS 2025 poster ()
Keywords: Tubular manifold
SE(3)-equivariant network，Graph neural network
Vessel classification
Primary area: deep_learning

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.

Topology-Aware Learning of Tubular Manifolds via SE(3)-Eqivvrrian Network on Ball B-Spline Curve
Jingxuan Wang! Zhongke Wu
Xingce Wang Zeyao Zhang'
Chunghao Zheng' Di Wang*
School of Artificial Intelligence, Beijing Normal University, China LLILY Research Centre, Nanyang Technological University, Singapore
Abstract
Tubular-like system shape analysis is quite difficull in geometry and topology, while it is widely used in plants and organs analysis in practice. However, traditional discrete representations such 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 (B33))BBSfooomeGGCC work 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 formal mathematical definition and analysis of the resulting mani- folds 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-bar baasedd 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 classifi- cation performance, generalization ability, convergence speed, and robustness overfitting.
novel frame-
Introduction
Tubular structures play a vital role in a wide range of tasks across computer vision, computer graphics, and interdisciplinary domains, including vascular segmentation [.]. classification (2), and traffic network planning [3] 44. However, the analysis of tubular-like models is quite difficult in both geometry and topology. From a geometric perspective, the branch morphology within tubular-like models varies significantly, with marked differences in tortuosity among individuals for branches of the same designation. From topological perspective, the morphology and connectivity of the branches are not unique. Certain branches are prone to stenosis, absence, or duplication, and tubular- like model structures frequently exhibit fusion crossover. Meanwhile, both the geometric structure and topological configuration of tubular-like models, such as vascular networks, may vary across different subjects even between different scans of the same subject.
The B-Spline model [ 6. 7. along with its extensions such as Ball B-Spline Curves (BBSC) S serves staustry- tool widely adopted in computer graphics and computer-aided design (CAD). In this work, we propose a compact and differentiable representation for tubular structures based BBSC, using small set of control points, radii, and knot vectors. Compared to existing voxel-based methods [ and point cloud- -ased methods (10,, our model not only reduces storage
FFrst author. Email: 20243108 1034 masl.mailbmi bmw.edu.cn i Conresponding author. Email: wangxingce@@ sangxingee@bnu.elu.cco bnu.edu.cn
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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overhead, but also enables accurate and interpretable geometric encoding. To enhance interpretability. we further define a smooth manifold structure over BBSC. To mitigate the sensitivity of geometric feature extraction to rigid transformations, we adopt a group-equivariant approach [1]] and propose SE(3)-BBSCformet a model that operates directly on the BBSC manifold and is equivariant under rotations and translations. By incorporating -E(3)--quivariancel [1], our method ensures consistent feature extraction under arbitrary rotations and translations, leading to improved stability, accuracy, and generalization in tubular structure analysis.
To model the braer- topological connectivity, we use Graph Convolutional Networks (GCNs) [13] as the predominant approach for learning topological features. Building on this foundation, we innovatively propose tubular manifold representation based on BBSC and construct topology- aware -quuivequiaa network (B1C))BBBCCoomrGCYY on this manifold. Our main contributions are as follows: 1) We define a compact and interpretable tubular manifold based on BBSC, construct its associated functional space M, and propose metric on M. An -qqiivaauv mapping, which enhances geometric consistency and symmetry preservation, constructed from M to a ligher-dimensionll functional space S, forming the architecture of SE(3)-BBSClormer 2) We define a topologically glued structure composed of multiple BBSC manifolds and design a corresponding GCN to model the topological relationships between manifolds, which plays critical role in handling structures with significant topological heterogeneity. 3) We conduct experiments on a multi-class classification task of the Circle of Willis (CoW) [14 [5]. Our proposed (B13))-BSCfoomerGCC demonstrates superior performance against voxel and point cloud state-of state-of-the-ar t--thhe-ttttt (SOTA) base- lines lines-in in terms of accuracy, computational efficiency, and generalization ability-on both public dataset and a clinical dataset collected in collaboration with a medical institution.
2
Related Work
Tubular structure extraction. Deep learning has been widely applied to tubular structure analysis
[16] DSCNet [17] captures local vessel tortuosity via dynamic snake convolution, while DUNet
integrates deformable convolutions into UNet [19] for retinal vessel segmentation. DDT [20]
employs distance transform to enable geometry-aware voxel segmentation. Topological constraints have also been proven essential. GraphMorph (2 and TopoLab [22) explicitly model topological connections improve performance
complex tubular structures. Meanwhile, TaG-Net (33
combines PointNet [24) with GNN (55 to extract both geometric and topological features for more accurate classification. The shape graph (26) models curves as edges and their intersections nodes, enabling mathematical analysis of graph shape variations. Moreover, it introduces a multi-scale representation to simplify the expression of complex graphs. However, all these methods rely on discrete representations- -ixeess voxels, or point clouds. In contrast, we propose a differentiable and compact representation using BBSC, enabling tubular structures to be modeled as manifolds with faithful geometric and topological expressiveness. Specifically, we represent tubular branches as BBSC- based nodes, while the connections between branches are treated as edges. This design avoids the complex process of defining edge weights and instead focuses on capturing the geometric characteristics of individual branches.
Group equivariant network. Group-equivariant models 11 12 have become central in geometric deep learning (27) due to their robustness and interpretability. Group-equivariant Convolutional Networks (G-CNNs) (8 first introduced group symmetry into CNNs, showing strong performance on symmetry-rich imaging tasks. This idea has since been extended to 3D shape feature extraction, where models like Tensor Field Networks (TFN) (29), (r(t))-Trnsformer [30] incorporate SE(3)- cquivariance into point cloud processing, significantly boosting both performance and stability. EquiTrrack (31) achieves equivariance via steerable CNNs to process temporal sequences. SpaER 332 propose the construction of equivariant spatial means using steerable CNN filters and introduced an innovative use of self attention mechanisms to process temporal sequences. MPerformer [3 employs an Srans--Trasformmer for molecular perception. However, the potential of applying group- equivariant models for geometric feature extraction in tubular structures has not been widely explored. To the best of our knowledge, we are the first to construct a ropology-aware and group-equivariant network on manifolds and apply it to the shape analysis and feature extraction of tubular structures.

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3
Preliminaries
Ball B-Spline Curves (BBSCs). BBSC an extension of B-Spline curves to spheres, with each sphere defined by a control radius. This formulation provides smooth representation for tubular objects with varying thickness. Given a set of control points {P) Pe..... Pr control radii {T1, 2,,..., rnj.
and a set of basis functions (t)) which are computed by a knot vector f1,,t2 ... Intpt1). BBSC of degree p is defined follows:
a Σ (t)
BBSC(t) = Σ (t,C
Σ N (t) 2
Σ (on)(t)) (I) 2m0
2
1 if Ststt t++1, and the higher degree
The basis function of degree 0 is initialized as No.
(o otherwise.
basis function (t)((t) can be calculated by iteratively interpolating the knot vector t, ......... f++p+1 and Ni.p-((t,, N1+1,p-1(t) as follows:
t tutp" 1tt- + 1+pp11 -t
(t)
Not2,p-
(2)
t+tp1 t,+1
BBSC consists of two parts: the center B-Spline curve (()) E (t,(()) and the radius function ott) Σi N (t)r,, Because 7(t) and (t) are calculated using the same basis function (Equation T and Figure D. they inherently inherit the mathematical properties of B-Spline curves, including differentiability, local control, the convex hull property, and favorable topological char- acteristics. More importantly, compared to voxels and point clouds, the BBSC representation can continuously and differentiably represent tubular object with a compact of parameters, which not only reduces storage and computational complexity, but also enhances theoretical interpretability and enables accurate geometric and analytical modeling.
n r(1) = Σ (1)(
dt) Σ (t)
BBSC(() (c(f); r(1)
1-
Figure 1: An example of BBSC. A. The 3D B-Spline curve is represented as linear combination of control points (red balls) and their associated basis functions. B. The tubular surface is formed by sweeping balls (yellow balls) parameterized by control radii along the B- B-Spline curve. C. The BBSC providing smooth and flexible curve representation in 3D Fuclidean space.
Group Representation. Group 34 is a concept from abstract algebra 35 used to describe symmetries and related properties. group consists set equipped with a binary operation and satisfies the following properties: closure, associativity and the existence of an identity element and an inverse element. The representation a group G GL(V), C R which satisfies V/91, 92 E G, p(g1 922 p(g1) ) p(gz). GL(V) denotes the set of all invertible linear transformations on V and p(g) corresponds to the matrix representation of group element g. The abstract group operation multiplication.
a homomorphic mapping (36) p:
be realized the corresponding matrix
Group Equivariance. A function to X + y is called G-equivariant if Vgg E G, given set of transformation (g) : X X, 3 transformation py(g) y -> ), such that
p(px(g)x) = p)()) x X
Equation can be simplified as (g)) = ((x) when the matrix representation of Py(g) is an identity matrix T. this case, &p is called invariant. If the function y defined in a Euclidean space I and it is equivariant under both rotation and translation transformations in this space, the function is referred 3D ato-trannlltion equivariant (Suiaiaqui,aariant) the group thus denoted SE(3).
3

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(e(3)-Transformer: The (E(3)-Transforme [30 is deep learning- based method that applies
group-equivariance auxiliary geometric features on 3D point clouds. constructs a set of SE(3)- equivariant basis kernels W
using spherical harmonics [37,, corresponding Clebsch-Gordan coefficients [38) and a learnable radial function. These are then integrated with an attention mechanism
39 form a group-equivariant network (see details in Appendix B.2). Given
an input Jin with channels, the output Fout with channels is obtained as follows:
fout W f +
ΣΣ
E JEN,
an, W (x; Iz).
(4)
where the attention matrix is computed as follows:
exp( ki.s) EyeN; exp(1, kis)
Σ We Jin kis
A W (2: f Oij
(5)
The direct sum + represents the concatenation of vectors. The attention mechanism is proven to be -equi--qiivaiian [30], as it relies on the inner product of ai and W (xz Zz), which is an SE(3)-equivariant operation.
4 Method and Analysis
Our method consists of two main components. First, we construct smooth BBSCs from discrete 3D data and, from a manifold perspective, define and analyze both their properties and the space they form. Second, we extend the for learning on continuous structures. We further employ a GCN capture topological relations and design pipeline (Figure 2 for branch classification in the CoW, tubular arterial network with specific topological connections.
to this manifold, yielding the
4.1 BBSC Functional Space and SE(3) Equivariant Mapping
We define the p-degree BBSC functional space Mp with given P, T, T from their respective vector spaces as follows:
Definition 1.
Mp = (3(P,r,)) () == (PE r Ry0: T TE.
(6)
where T = ftz E [0. 1 t .... 1 E [0. 1] and B(P,r, r), r) is the total structure of BBSC (see Equation (I), which is dinediina manifold. In order to further stabilize BBSC, we impose certain constraints on T (t, E [0, [0.1]]t++1 < ftp2 < ... < t, =
tp+1 = entries of the knot vector are set to 0 and 1,
1ntp+1 1}. The first and last respectively, to ensure that the two endpoints of the curve coincide with the first and last control points. The parameter f in Equation ① specifies positions along the curve but does not affect the overall BBSC shape, serving only as an internal parameter independent of B E .M. Because 3(P.r, r..r. is linear combination P and r, it has infinitely many continuous partial derivatives. By excluding the first and last + 1 entries of r, i.e., in the open subset of M, that omits two boundary points, (t) becomes a recursive fractional linear function of т, and differentiable with respect of arbitrary order. Moreover, under the constraint that the middle entries of are strictly monotonically increasing, 3(P, r) possesses infinitely many continuous partial derivatives. According Equation (I). P. r. and r are in a separable product form. Consequently, their higher-order mixed partial derivatives can be expressed as products of higher-order partial derivatives, which remain continuous. Therefore, Mp can be regarded as ( (ox(O, (0,Mp) Mp). The complete proof is provided in Appendix C.
0,
The metric on MD can be simply defined weighted sum of the L2 norms of the parameters as follows:
g = - P||2 + a - Fll2 + 7/|T - Fll2-
(7)
Although the metric based on L.2 norms provides a simple way to measure parameter variations, ignores the perturbations independent (hence P and r are independent). whereas 7 simultaneously influences both. Thus, the coupling between and P, should be considered when defining the metric. To address this,
Mp. Moreover, 7()) and (l), which determine the BBSC shape, are

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propose a metric that accounts for perturbations on Mp and parameter coupling. Concretely, we first characterize perturbations in the tangent space of M, as by and åo as follows:
87 (t)pt) + EEP, (N dT, do Σ Nip(t)on
+ Σ ON.,(t)
P. δT
δT.
(8)
i=1
==1
Given geometric perturbations 83 (6y, and 83 = (ō%, 80) in the tangent space of Mps we define the following inner product follows:
10:0:M,/.. (53,58) = -fo0 (167).61 mx
067 0b7, adodo + ar ot 0t w(t)dtt 0t Ot
dōa Oōo
(9)
where a 2 0, 77,, no 2 O, and w()) denotes a weighting function. One may simply set (() = to perform unweighted integration. However, recommend using w(t) (t)]l, which corresponds to arc-length weighting, thereby mitigating the effect of reparameterization and enhancing geometric interpretability. The proposed inner product is inspired by the Sobolev H² inner product 440 41],
which incorporates only the Euclidean inner product of (ōny, .0) and (ō7, 80)) themselves, but also their first-order derivatives. Because (.,.) denotes the Euclidean inner product, it inherently possesses rotation and translation invariance. Consequently, (,-),M, inherits these desirable geomet- ric properties. Moreover, this inner product preserves the separability of parameters P and r, while simultaneously coupling them with т through Equation (8) Let (P, r) and 50 (6P, ör; år).
Denote the Jacobian of by D3( follows:
89
Then, the pullback metric can be finally written as
86
g (50, e (53, M, (D3(0)50, (B(0)88) M2
(10)
Similarly, we can define the entire BBSC functional space M as follows.
Definition .M is called BBSC functional space if it contains all existing Mp, as denoted as M = Up20 Mp.
Definition 3. A mapping : :Mp - S is deemed SF(3) on Mp. (B( E Mp and V transformations g E SE(3), the following equation holds:
(0)( r)) ps ps(())p((P,r,.7)
(11)
op infinitely differentiable with respect to P, r, and т, yp is C mapping on MM, meaning smooth as well.
Definition G.M a manifold with topological relations constructed from the direct sum of disjoint BBSC manifolds 31, B2,... 3x E M, which is defined as follows:
GM
L 3/ / ~ ~ Vel
L 3, SET
x
L 3 iEl
(12)
Specifically, G.M is a topological manifold composed of multiple M with certain topological relationships between them. GM can be viewed manifold with gluing structure [42 43 44
4.2 BBSC Manifold Construction
We follow [55 and adopt a similar BBSC fitting algorithm. Given set of points V E Nx3 from tubular objects, we apply DiffusionNet [46] (details in Appendix B. to embed them and output BBSC, represented by control points
(Pi, Pz..... Pr), control radii r {ri, F2, r,. ) and To ensure 1p+2 < 4p+ < < 1 and 4 =
a knot vector (4, 12,,..... ... 1. 0, ln ....... lutp+1 = 1, we predict the differences of the inner knot vector VT using softmax function, and then obtain the full knot vector by applying cumulative sum and concatenating the boundary zeros and ones. Considering all possible control parameter sets P E gaxn and r E R", together with constrained knot vectors smooth mapping B E (p)
1p+1
construct the functional space M, endowed with
We introduce an additional template spline 3 to constrain the shape of the BBSC centerline. We minimize a loss function composed of two terms, the Hausdorff distance Dæ [47 between sampled
5

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points on the constructed BBSC 3 and the raw surface points, and the metric between 3 and B. Note that the Hausdorff distance can be replaced by the Chamfer distance 48], and a regularization term Jo x² (t) dt, which can be used to enforce the global smoothness of the BBSC centerline and also be incorporated into the loss function. Based these, further incorporate topological relationships among the 3 to construct the topological manifold G.M.
4.3 SE(3)-BBCCformer and Graph Convolutional Networks
pusta c6Ba
SEC3I BBSCformer
SEISI Equive
Mapping
Control Points
Saher Harmonics
Control Radius
MLP
Knot Vector
MLP
CG Confficietss
o Messagge Passing
Figure 2: The BBC) takes vascular centerlines and surface point clouds inputs. fitting module is first employed to construct BBSC. These spline-based manifold representations then mapped into high-diimensional manifold space via the SE(3)-BBSCiormer Finally, GCN is utilized to process the topological relationships among the manifolds.
SE(3)- Transforrno can be viewed as a mapping operating in a discrete space represented by a point cloud. In contrast, the BBSC model is smooth and continuous representation. To integrate the advantages both, we reconstruct the model and term it SE(3)- BBSCClrree,,, which can interpreted as an SE(3) mapping defined on M. First, we compute the centroid of the control points 7 = >m to serve as the virtual anchoring point, acting as a central hub for transmitting information between the control points and generating the unit vector Pt P--P [P: needed by SE(3)-BBSCfomer We initialize
the Ci channel feature f combination of vectors generated from and follows:
f = (MLP(r), MLP(f)), 7 ((t1.....)))........... ((t2....,f fp+1)........................... (tn, ... (n+p-1) E Rnx(p+1),
Then we view each control point P; as being connected with P and replace the x; and x, with P; and 7 in Equation 4 and (5) The resulting SE(3)-BBSCforme is defined as follows:
fout, = Wor fir aa. -P)fin
(13)
where the attention matrix is computed as follows:
explg.Tk
T D Weoti for kj
Σ wes - -P) ag
(14)
Σ exp(7."1 ky)
For convenience, we abbreviate -B())-orBSfoomer as the mapping P MM + S, where S denotes the high-dimensional functional space. Variable sp(B can be interpreted as the high-dimensional representation of BBSC in S. If an object is a manifold with topological relationships, composed of multiple B components as described Definition 4, then GCN can be applied capture the topological relationships. Each BBSC manifold B G.M is treated node in the graph G and

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(((5 serves the initial node feature v. For simplicity, we initialize the edge attribute e as the difference between features of the connected node features. Then, the GCN is formulated as follows:
MLP (1yg, en,))) ))
Eij = MLP([ ej, etyD. e, = MLP
Σ JEN;
(15)
Gill
Taking the CoW as an input example, we propose the (BL))-BBSCFOmerGGNN, The pipeline of our model is shown in Figure
4.4 Theoretical Analysis and Discussion of Advantages
The BBSC mainfold representation / is highly compact, allowing complex tubular shapes to be described using only small number of control parameters and knot vector, thereby substantially reducing storage and computational costs. Moreover, the BBSC manifold exhibits strong control- lability because its shape can be easily adjusted by modifying the control parameters and the knot vector. Due to the smoothness of B. geometric and differential quantities such as normals, curvature, and torsion can be accurately calculated at any position in the object, without relying on numerical approximations that are prone to errors when estimated from discrete data 49 55 This property
endows the representation with stronger expressive power and improved theoretical interpretability. Moreover, 3 can be shown to admit continuous higher-order partial and mixed derivatives for all parameters, i.e., 8 E C This ensures that M forms a continuous and infinitely differentiable functional space. Benefiting from these properties, the BBSC manifold representation is particularly well-suited for modeling complex geometric tubular structures.
The SE(3) -E(3)-equivariant mapping : .M- S can be shown be a C mapping. Spherical harmonics are also (* [5]]. so the differentiability of ip primarily depends on that of the MLP. In turn, the MLP's differentiability determined by the choice of activation function. In this work, all MLPs are constructed using GELU activations and linear layers, both of which are O mappings. Consequently, & C mapping from M S that satisfies SE(3) equivariance. The group- equivariant &p preserves the intrinsic symmetries of M when mapping to S, ensuring the stability of its shape features. Moreover, the GCN designed to capture topological relationships maintains the overall topological stability of the manifold space, thereby enhancing the model's generalization and robustness while remaining lightweight.
5 Experiments and Analysis
We evaluate the performance of our proposed model on the branch classification task for the CoW, critical arterial structure located at the base of the human brain. Experiments are conducted on two datasets: the TopCoW dataset from the MICCAI 2024 Challenge (52), which contains 125 samples across 13 anatomical classes, and clinical dataset collected from collaborating medical institution, comprising 1,182 samples across 22 classes. We focus on the CoW branch classification
task because it is complex tubular structure characterized by both geometrie and topological properties, and necessitate efficient computational approaches. Moreover, CoW exhibits highly bilateral symmetry while simultancously presenting complex topological heterogeneity, making group equivariant networks and GCNs particularly well-suited for modeling such structures. (see detailed descriptions of the CoW datasets, preprocess and training protocols in D)
5.1 Experiments on TopCoW 2024 MICCAI Challenge
We evaluate our model on the publicly available TopCoW dataset (52]. We include several SOTA voxel-based and point cloud-based classification methods baselines. In addition, we conduct ablation studies to investigate the effects of incorporating SE(3)-equivarianl BBSC mapping and the use of GCN for modeling topological relationships. Given the small sample size of the TopCow 2024 dataset, we employ 5-fold cross-validation to enhance training robustness. We calculate the mean and standard deviation, highlighting the highest mean and lowest standard deviation in bold. Our model achieves the highest mean and lowest variance across AUC-ROC, Recall, and Fl score, while also demonstrating competitive performance in Accuracy and Precision (see results in Table []
Across the major- We illustrates the class- wise prediction accuracy on the CoW dataset (see Figure 3 ity of labels, our model consistently outperforms baseline methods. Due to the highly heterogeneous
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Table 1: Performance comparison of different methods on the TopCoW 2024 dataset
Method
Accuracy %
AUC-ROC%
Precision %
Recall %
Fl Score %
3D DenseNet 53
97.02 0.57
99.63
1.94
95.86 = 0.51
95.42 + 0.41
97.47 + 0.48
3D ResNet 54
96.54 0.54
99.61 0.11
96.79 = 0.76
96.44 = 0.83
96.52 + 0.80
PointNet 24
94.10 0.88
0.20 99.46
94.81 = 1.36
94.69 = 0.92
94.57 = 1.10
PointNet++ [55
96.61 = 0.53
99.61
0.09
96.72 + 0.49
96.52 = 0.55
96.45 + 0.60
CurveNet [56]
97.14 1.16
99.72 0.20
97.31 1.16
97.07 = 1.19
97.09 - 1.22
RepSurf-UU [57
97.41 0.68
99.86
0.07
96.83 0.52
96.61 0.54
96.50 + 0.55
PointMLP [58
94.10 2.27
99.03 0.56
94.41 2.35
94.05 = 2.31
93.94 + 2.40
Transforreer 30
61.35 5.80
90.93 1.31
62.36 5.28
60.77 = 5.70
60.58 + 5.69
87.91 1.35
98.24
0.31
88.97 = 1.66
88.01 = 1.32
87.98 + 1.26
BBSCformerGCN
97.95 + 1.22
99.89 0.02
97.25 = 0.79
98.34 = 0.60
97.78 - 0.70
97.53 0.83
99.99 0.01
97.36 - 0.53
98.44 = 0.31
97.87 = 0.41
Label Accuracy Distribution
Pretie
= Curveler
= Neyikatiu
1.0
Accuracy
0.6
0.4
0.2
10
12
Label
Figure 3: Per-class prediction accuracy of various models evaluated on TopCoW 2024 dataset.
anatomical structure of the CoW 55 60) 61, the distribution of different branches label is notably imbalanced. For example, in the TopCoW 2024dataset, only 5% of the 125 samples contain the segment corresponding to Label 12 (see details in Appendix D.1). This extreme scarcity introduces a high degree of randomness during testing. However, we observe that this issue is effectively mitigated by increasing the total number of training samples, as further demonstrated in Section 5.2.
5.2 Experiments on Real-World Clinical Data
Table 2: Performance comparison of different methods on the clinical dataset
Method
Accuracy%
AUC-ROC%
Precision%
Recall%
Score
PointNet 24
69.20 + 0.98
96.50 = 0.24
70.02 1 0.77
69.17 = 0.96
68.90 +: 1.02
PointNet++
551
70.27 + 0.16
96,60 = 0.17
70.82 + 0.43
70.28 = 0.20
70.05 +:20 0.14
CurveNet
[56]
81.91 + 0.14
93.80 + 1.13
62.92 + 3.14
60.13 + 4.25
58.51 + 4.58
RepSurf-U
[57
78.22 0.17
98.40 0.11
78.34 + 0.28
77.99 = 0.29
78.00 + 0.25
PointMLP
[58
82.70 +:700:00 0.10
98.66 = 0.06
82.86
+ 0.14
82.70 = 0.10
82.67
+ 0.14
SE(3)-Transformer
30
61.96 + 1.45
92.32 0.96
61.05 + 2.07
61.91 = 1.47
60.77 +: 2.57
SH(3)-BBSCformer
71.06 + 1.39
93.25 + 0.60
70.25 + 1.16
71.04 + 1.39
70.45 + 1.21
BBSCformerGCN
95.45
1.09
99.34
0.21
95.10 + 1.48
95.38 + 1.13
95.23 + 1.29
SB(3) BBSCforeCNN
96.11 + 1.01
99.38 + 0.19
96.11 + 0.93
96.08 + 0.97
96.02 + 1.05
We further validate the generalization ability of our model on clinical data collected from the collaborating medical institution. This dataset contains approximately ten times more number of samples than the TopCoW 2024 MICCAI Challenge dataset. Although the inherent heterogeneity
8

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of CoW still leads to imbalanced class distributions, the substantially larger sample size helps alleviate the randomness predictions for underrepresented classes. The results presented in Table [ demonstrate that our model exhibits stronger generalization ability compared to voxel-based and point-based approaches. While other methods experience a performance drop of 10-20%, our model maintains consistently high level of evaluation metrics. Notably, SH(3)- BBSCformeeGGCN outperforms BBSCformerGCN in both mean and standard deviation across all evaluation metrics, further validating the stability and generalization advantages introduced by the SE(3)-equivariant mapping. Additionally, the comparison between SEC3)-BBCCormerGCNN and highlights the critical role of topological information. Collectively, these findings suggest that both geometric and topological stability are the key factors underpinning the generalization capability of our model. Similarly, we provide the per-class prediction accuracy across all label categories As shown, our model consistently outperforms all baselines across all 22 classes
(Figure 4).
Label Accuracy Distribution
SE1S:BBSCCeemeCON
Painther
Peinshe2
PainMUL
Curvelt:
RegSurfAJ
10-
08-
Accuracy
8
2
23
14
35
28
39
20
Label
Figure 4: Per-class prediction accuracy of various models evaluated on real-world clinical dataset.
5.3 Training Efficiency, Stability, and Computational Cost Analysis
We illustrates the training and testing accuracy curves on the clinical datasets (see Figure 5) Our method consistently achieves faster convergence and greater stability across all methods. These advantages are largely attributed to the compact input representation, which preserves fine-grained geometric details without the need for aggressive downsampling. More critically. while other models exhibit severe overfitting, both strong resistance to overfitting, highlighting the robustness and generalization power of equivariant architectures. Moreover, leveraging BBSC's to capture complex geometric features and topological structures of tubular systems further enhances our model's 's performance over existing baselines.
and the 3E(3)-Transforms demonstrate
Clinical Training Accuracy
Clinical Test Accuracy
1.0
1.0
0.9
0.9
0.7
0.7
0.6
SEI3I BBSCTomennGc
0.4
PontMLP
0.4
PotntMLP
CurveNte:
CurveNet
Wansformer
SE/31 Transformrr
PontNet
Painthet
61
PontNe12
01
Pointhet2
0
25
75
100
125
150
175
200
23
50
75
1000
125
150
175
200
Esoch
Epoch
Figure 5: Epoch-wise training and testing accuracy evaluated the clinical dataset.
To validate the efficiency of BBSC for tubular object representation, we compared our model against several basclines evaluation time, FLOPs, and parameter count (see Table 3) SE(3)-

---

BBSCformerGCN achieves superior generalization and classification performance while significantly reducing inference time and memory usage. These results underscore the practical benefits of our design, particularly for fast and cost-effective deployment in clinical settings.
Table 3: Parameters amount and computing cost comparison of different methods
Method
Evaluation Time(ms)
FL.OPs(M)
Parameters(M)
3D DenseNet
53
8.03
9743.76
18.56
3D ResNet 54
10.95
11463.08
85.23
PointNet 24
0.88
450.38
3.46
PointNet++
55
1.54
4067.53
1.74
CurveNet [56
81.95
269.56
2.13
RepSurf-U [57
59.84
911.32
1.48
PointMLP 58
38.39
15733.95
13.23
SB(3)-Transe Transforrrr
19.03
456.33
0.12
SE(3)-BBSCfomg
1.92
1.72
0.35
BBSCCoreerGCN
1.14
88.98
3.60
BBLD)-BBSCCoomrrGGN
3.02
59.01
2.70
6 Discussion and Future Work
We leverage an extension of B-spline curves to the spherical domain, which constructs the BBSC, to model tubular structures. We formulate mathematically well-defined smooth manifold based on this representation, construct a functional space for tubular manifold, and discuss the associated metric on this space. Furthermore, we propose rotation and translation equivariant mapping, which applied to the anatomical classification of the CoW. While our experiments focus on this clinically significant cerebrovaseular structure, the proposed framework is readily applicable to wide range of other tubular structures. Additionally, the BBSC structure is naturally extensible. The control vectors be lifted into an n-dimensional space, where the B-spline basis functions can be reused to construct smooth and high-dimensional manifolds. Notably, SplineCNN [62 has previously utilized this insight define smooth convolutional filters via B-Spline kernels. Lastly, our model maps M defined by BBSC into siigh-dimessinall space S, preserving parameters P. However, a limitation our current model is that the graph convolution operations used to model topological relations on S are not equivariant, which could compromise the overall stability and full equivariance of the model. worth investigating into the design of equivariant mappings in dime-dima topological manifold spaces as line of future work.
of the 3D
Conclusion
We propose SE(3)-BBSCCommeGGCN novel deep learning architecture that achieves group equiv- ariance on tubular manifolds. Our model takes as input BBSC manifold, parameterized by a small set of 4D control parameters and knot vectors. This representation not only enables more effective extraction of complex geometric and topological features from tubular structures, but also reduces computational complexity. This design offers strong mathematical interpretability and improved computational efficiency. We validate our model on a clinically important yet computationally underexplored task: classifying anatomical configurations of the CoW, and demonstrate its superior performance in classification,, generalization, and resistance to overfitting. Our code will be available on
8 Acknowledgments
This research partially supported by the National Natural Science Foundation of China (Grant No. 62072045), the Beijing Municipal Science and Technology Commission and the Zhongguancun Sci- ence Park Management Committee (Grant No. Z221 100002722020), of Beijing (Grant No. 7242167), and the Teaching Reform Project of Beijing Normal University. We would also like to express our sincere gratitude to Peking Union Medical College Hospital (PUMCH) for kindly providing the clinical CoW scan data used in this study.
Natural Science Foundation
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References
[1] Joes Staal, Michael D Abråmoff. Meindert Niemeijer, Max A Viergever, and Bram Van Ginneken, Ridge- based vessel segmentation color images the retina. IEEE transactions on medical imaging, 23(4):501- 509, 2004.
[2] Volodymyr Mnih. Machine learning for aerial image labeling. University of Toronto (Canada), 2013.
[3] Volodymyr Mnih and Geoffrey E Hinton. Learning to detect roads in high-resolution aerial images. In Computer Vision-ECCC 2010: 11th European Conference on Computer Vision, Heraklion, Crete, Greece, September 5-11, 2010, Proceedings, Part VI 11, pages 210-223. Springer, 2010.
[4] Luisa Costa Sousa, Catarina F Castro, Carlos Conccição António, and Rui Chaves. Blood flow simulation and vascular reconstruction. Journal of biomechanics, 45(15):2549-2555 2012.
[5] Carl De Boor. A practical guide to splines, volume 27. springer New York, 1978.
Carl De Boor. On calculating with b-splines. Journal of Approximation theory, 6(1):50-62, 1972.
Michael Unser, Akram Aldroubi, and Murray Eden. B-spline signal processing. theory. IEFE transactions on signal processing, 41 (1)):221-833 1993.
[8] Zhongke Wu, Hock Soon Seah, and Mingquan Zhou. Skeleton based parametric solid models: Ball b-spline curves. In 2007 10th IEEE International Conference on computer-aided design and computer graphics, pages 421-424. IEEE, 2007.
[9] Rafic Nader, Romain Bourcier, and Florent Autrusseau. Using deep learning for an automatic detection and classification of the vascular bifurcations along the circle of willis. Medical image analysis, 89:102919, 2023.
[10] Dong Wang. Zhao Zhang, Ziwei Zhao, Yuhang Liu, Yihong Chen, and Liwei Wang. Pointscatter: Point set represcntation for tubular structure extraction. In European conference on computer vision, pages 366-383. Springer, 2022.
[11] Jan E Gerken, Jimmy Aronsson, Oscar Carlsson, Hampus Linander, Fredrik Ohlsson, Christoffer Petersson, and Daniel Persson. Geometric deep learning and equivariant neural networks, Artificial Intelligence Review, 56 14605--666655 2023.
[12] Carlos Esteves. Theoretical aspects of group equivariant neural networks, arXiv preprint 154, 2020.
[13] Thomas N Kipf and Max Welling. supersupe classification with graph convolutional networks. arXiv preprint arXiv:1609.02000 0060900907.21 2016.
[14] Bernard Alpers, Richard G Berry, and Richard M Paddison. Anatomical studies of the circle of willis in normal brain. AMA Archives of Neurology & Psychiatry, 81(4):409 418, 1959.
[15] Sumaiya lgbal. A comprehensive study of the anatomical variations of the circle of willis in adult human brains. Journal clinical and diagnostic research: JCDR, 7(11):2423, 2013.
[16] Jie Mci, Rou-Jing Li, Wang Gao, and Ming-Ming Cheng. Coanet: Connectivity attention network for road extraction from satellite imagery. IEEE Transactions on Image Processing, 30:8540 8552, 2021.
[17] Yaolci Qi, Yuting He, Xiaoming Qi, Yuan Zhang, and Guanyu Yang. Dynamic snake convolution based on topological geometric constraints for tubular structure segmentation. In Proceedings of the IEEE/CVF international conference on computer vision, pages 6070-6079, 2023.
[18] Qiangguo Jin, Zhaopeng Meng, Tuan D Pham, Qi Chen, Leyi Wei, and Ran Su. Dunet: A deformable network for retinal vessel segmentation. Knowledge-Band Based Systems, 178:149-162. 2019.
[19] Olaf Ronneherger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In Medical image computing and computer-asiisted intervention-MICCAA 2015: 18th international conference, Munich. Germany, October 5-9, 2015, proceedings, part III 18, pages 234-241. Springer, 2015.
[20] Yan Wang. Xu Wei, Fengze Liu, Jieneng Chen, Yuyin Zhou, Wei Shen, Elliot Fishman, and Alan Yuille. Deep distance transform for tubular structure segmentation in ct scans. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pages
2020.
11

---

[21] Zhao Zhang, Ziwei Zhao, Dong Wang, and Liwei Wang. Graphmorph: Tubular structure extraction by morphing predicted graphs. Advances Neural Information Processing Systems, 37:68472-68499, 2024.
[22] Zhixing Zhang, Ziwei Zhao, Dong Wang. Shishuang Zhao, Yuhang Liu, Jia Liu, and Liwei Wang. Topology- preserving automatic labeling of coronary arteries via anatomy -wwae connection classifier. In International conference on medical image computing and computer- assisted intervention, pages 759-769. Springer, 2023.
[23] Linlin Yao, Feng Shi, Sheng Wang, Xiao Zhang, Zhong Xue, Xiaohuan Cao, Yiqiang Zhan, Lizhou Chen, Yuntian Chen, Bin Song. et al. Tag-net: topology-awan aware graph network for centerline baaed vessel labeling. IEEE transactions on medical imaging, 42:11)) 11:1133333-116 2023.
[24] Charles R Qi. Hao Su. Kaichun Mo, and Leonidas J Guibas. Pointnet: Deep learning on point sets for 3d classification and segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 652-660, 2017.
[25] Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S Yu. A compre- hensive survey graph neural networks. IEEE transactions on neural networks and learning systems. 3241)),424, 2020.
[26] Aditi Basu Bal, Xiaoyang Guo, Tom Needham, and Anuj Srivastava. Statistical analysis of complex shape graphs. IEEE Transactions on Pattern Analysis and Machine Intelligence, 46(12))8788 8805, 2024.
[27] Jiaqi Han, Jiacheng Cen, Liming Wu, Zongzhao Li, Xiangzhe Kong, Rui Jiao, Ziyang Yu, Tingyang Xu, Fandi Wu, Zihe Wang, et al. A survey of geometrie graph neural networks: Data structures, models and applications. arXiv preprint arXiv:2403 00485, 2024.
[28] Taco Cohen and Max Welling. Group equivariant convolutional networks. International conference on machine learning, pages 2990-2999. PMLR, 2016.
[29] Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick Riley. Tensor field networks: Rotation-an and translation translatio-cquiivaiiatt neural networks for 3d point clouds. arXiv preprint arXiv: 0888000829999 2018.
[30] Fabian Fuchs, Daniel Worrall, Volker Fischer, and Max Welling, Se (3)-traasformers: 3d roto-tramslation cquivariant attention networks. Advances in neural information processing systems, 33:1970- 2020.
[31] Benjamin Billot, Neel Dey, Daniel Moyer, Malte Hoffmann, Esra Abaci Turk. Borjan Gagoski, Ellen Grant, and Polina Golland. Se equvaiarari and noise-invarian 3d rigid motion tracking in brain mri. IEEE transactions on medical imaging, 43(1)44402 4040, 2024.
[32] Jian Wang, Razieh Faghihpirayesh, Polina Golland, and Ali Gholipour. Spaer: Learning spatio-temporal equivariant representations for fetal brain motion tracking. In International Workshop Preterm, Perinatal and Paediatric Image Analysis, pages 3 13. Springer, 2024.
[33] Fanmeng Wang, Hongteng Xu. Xi Chen, Shuqi Lu, Yuqing Deng, and Wenbing Huang. Mperformer: An se (3) transformer--asee molecular perceptron. In Proceedings of the 32nd ACM International Conference on Information and Knowledge Management, pages 2512-2522, 2023.
[34] William Raymond Scott. Group theory. Courier Corporation, 2012.
[35] Joseph Gallian. Contemporary abstract algebra. Chapman and Hal//CRC, 2021.
[36] Ryszard Raczka and Asim Orhan Barut. Theory of group representations and applications. World Scientific Publishing Company, 1986.
[37] Claus Müller. Spherical harmonics, volume 17. Springer, 2006.
[38] Johan J de Swart. The octet model and its clehsch-gordan coefficients. In The Fightfold Way, pages 120-143. CRC Press, 2018.
[39] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017.
[40] Robert A Adams and John JF Fournier. Sobolev spaces, volume 140. Elsevier, 2003.
[41] Haim Brezis and Haim Brézis. Functional analysis, Sobolev spaces and partial differential equations. volume 2. Springer, 2011.
12

---

[42] Victor Guillemin and Alan Pollack. Differential topology, volume 370. American Mathematical Soc.. 2010.
[43] James R Munkres. Elements of algebraic topology. CRC press, 2018.
[44] Allen Hatcher. Algebraic topology. 2005.
[45] Zhongke Wu, Xingee Wang, Yan Fu, Junchen Shen, Qianqian Jiang, Yuanshuai Zhu, and Mingquan Zhou. Fitting scattered data points with ball b-spline curves using particle swarm optimization. Computers graphics, 12:1-11, 2018.
[46] Nicholas Sharp. Souhaib Attaiki, Keenan Crane, and Maks Ovsjanikov. Diffusionnet: Discretization agnostic learning on surfaces. ACM Transactions on Graphics (TOG), 41(3):1-16, 2022.
[47] Daniel P Huttenlocher, Gregory A. Klanderman, and William J Rucklidge. Comparing images using the hausdorff distance. IEEE Transactions on pattern analysis and machine intelligence, 15(9)850--63, 1993.
[48] Harry G Barrow, Jay M Tenenbaum, Robert C Bolles, and Helen C Wolf. Parametric correspondence and chamfer matching: Two new techniques for image matching. Technical report, 1977.
[49] Weihuan Chen, Shing--sen Chern and Kai S Lam. Lectures on differential geametry, volume 1. World Scientific Publishing Company, 1999.
[50] Richard L Bishop and Richard Crittenden. Geometry of manifolds: Geometry of Manifolds, volume 15. Academic press, 2011.
[51] George B. Arfken, Hans Weber, and Frank Harris. Mathematical Methods for Physicists: A Compre- hensive Guide. Academic Press, 2011.
[52] Kaiyuan Yang, Fabio Musio, Yihui Ma, Norman Juchler, Johannes C Pactzold, Rami Al-Maskari, Luciano Höher, Hongwei Bran Li, Ibrahim Ethem Hamamci, Anjany Sekuboyina, et al. Benchmarking the cow with the topcow challenge: Topology-aware anatomical segmentation of the circle of willis for cta and mra. 2024.
ArXir, pages
[53] Sen Liang. Rongguo Zhang, Dayang Liang. Tianci Song, Tao Ai, Chen Xia, Liming Xia, and Yan Wang. Multimodal 3d densenet for idh genotype prediction in gliomas. Genes, 9(8):382, 2018.
[54] Zhaofan Qiu, Ting Yao, and Tao Mei. Learning spatio-temporal representation with pseudo-3d residual networks. In proceedings of the IEEE International Conference on Computer Vision, pages 2017.
[55] Charles Ruizhongtai Qi, Li Yi, Hao Su, and Lconidas Guibas. Pointnct++: Deep hicrarchical feature learning on point sets in a metric space. Advances in neural information processing systems, 30, 2017.
[56] Tiange Xiang, Chaoyi Zhang. Yang Song, Jianhui Yu, and Weidong Cai. Walk in cloud: Learning curves for point clouds shape analysis. In Proceedings of the IEEE/CVF international conference on computer vision, pages 915-924, 2021.
[57] Haoxi Ran, Jun Liu, and Chengjie Wang. Surface representation for point clouds. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 18942-18952, 2022.
[58] Xu Ma, Can Qin, Haoxuan You, Haoxi Ran, and Yun Fu. Rethinking network design and local geometry in point cloud: A simple residual mip framework. arXiv preprint arXiv:2202.07122, 2022.
[59] James R Ayre, Peter J Bazira, Mohammed Abumattar, Haran N Makwana, and Katherine Sanders. new classification system for the anatomical variations of the human circle of willis: A systematic review. Journal of Anatomy, 240(6): 111-111874444 1204, 2022.
[60] Joshua Jones, Pedro Castanho, Peter Bazira, and Katherine Sanders. Anatomical variations of the eirele of willis and their prevalence, with focus on the posterior communicating artery: a literature review and meta-analysis. Climical Anatomy, 34(7):978-990, 2021.
[61] Lars B Hindenes, Tor Ingebrigtsen, Jorgen Isaksen, Asta K Håberg. Liv- Liv-Hege Johnsen, Marit Herder, Ellisiv Mathiesen, and Torgil Vangherg. Anatomical variations in the circle willis are associated with increased odds of intracranial ancurysms: The tromsø study. Journal of the Neurological Sciences, 452:1 452:120744 20740, 2023.
[62] Matthias Fey, Jan Eric Lenssen, Frank Weichert, and Heinrich Müller. Splinecnn: Fast geometric deep learning with continuous b-spline kernels. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 869 877, 2018
13

---

[63] Alvaro Banchezz-onzale.. Jonathan Godwin, Tobias Pfaff, Rex Ying, Jure Leskovec, and Peter Battaglia. Learning simulate complex physics with graph networks. In International conference on machine learning, pages 8459-8468. PMLR, 2020.
[64] Tobias Pfaff, Meire Fortunato, Alvaro Sanchez-Gonzzlerr, and Peter Battaglia. Learning mesh-based simulation with graph networks. International conference on learning representations, 2020.
[65] Oscar Méndez-Lucis Mazen Ahmad, Ehecadl Antonio del Rio-Chanona, and Jörg Kurt Wegner. geometric deep learning approach to predict binding conformations of bioactive molecules. Nature Machine Intelligence, 3(12):1033-11 1039, 2021.
[66] Jingxuan Wang, Zongzhao Qiu, Xuan Zhang, 7henghe Yang, Wei Zhao, and Xuefeng Cui. Boosting deep learning-based docking with cross-attention and centrality embedding. In 2022 IEFP International Conference on Bioinformatics and Biomedicine (BIBM), pages 360-365. IEEE, 2022.
[67] Michael M Bronstein, Joan Bruna, Taco Cohen, and Petar Veličković. Geometric deep learning: Grids, groups, graphs, geodesies, and gauges. arXiv preprint arXiv:2104.13475 2021.
[68] Federico Monti, Davide Boscaini, Jonathan Masci, Emanuele Rodola, Jan Svoboda, and Michael M Bronstein. Geometric deep learning on graphs and manifolds using mixture model cnns. Proceedings of the IEEE conference on compuder vision and pattern recognition, pages 5115-5124, 2017.
[69] Farhad Arbab, Ivan Herman, and Pal Spilling. An overview of manifold and its implementation. Concur- rency: practice and experience, 5(1): 1993.
[70] Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895, 2020.
[71] Lu Lu, Pengzhan Jin, and George Fm Karniadakis. Deeponet: Learning nonlinear operators for identi- fying differential equations based on the universal approximation theorem of operators. arXiv preprint arXiv: 191.0339,,,,93,, 2019.
[72] K Kapoor, VK Kak, and Singh. Morphology and comparative anatomy of circulus arteriosus cerebri in mammals. Anatomia, histologia, embryologia, 32(6):347 -555, 2003.
[73] Zvonimir Vrselja, Hrvoje Brkie, Stefan Mrdenovie, Radivoje Radic, and Goran Curic. Function of circle of willis. Journal of Cerebral Blood Flow & Metabolism, 34(4):578-584, 2014.
[74] Julie Rosner, Vamsi Reddy, and Forshing Lui. Neuroanatomy, circle of willis. 2018.
[75] Minghui Zhang, Xin You, Hanxiao Zhang, and Yun Gu. Topology-awar exploration of circle of willis for cta and mra: Segmentation, detection, and classification. arXiv preprint arXiv:2410./550 2024.
[76] Li Chen, Thomas Hatsukami, Jenq-Neng Hwang, and Chun Yuan. Automated intracranial artery labeling using graph neural network and hierarchical refinement. In Medical Image Computing and Computer Assisted Intervention MICCAI 2020: 23rd International Conference, Lima, Peru, October 4 8, 2020, Proceedings, Part 23. pages 76 Springer, 2020.
[77] Eugene Wigner. Group theory: and its application to the quantum mechanics of atomic spectra, volume 5. Elsevier, 2012.
[78] Lars B Hindenes, Asta Håberg, Liv Hege Johnsen, Ellisiv Mathiesen, David Robben. and Torgil Vangberg. Variations in the circle of willis in a large population sample using 3d tof angiography: The tromsø study. PL.S One, 15(1 1334443333, 2020.
[79] Monique Krabbe- Harthk-pp Berend Hillen, MM Breteler, and WP Mali. Circle of willis: morphologic variation on three-dimensional time-of-flight mr angiograms. Radiology, 203(1):033-11,, 1998.
Jeroen Van der Grond, Frank Erik De Leeuw, Jan Cees De Groot, Ale Algra,
[80] Mika Okahara, Hiro Kiyosue, Hiromu Mori, Shuichi Tanoue, Michihumi Sainou, and Hirohumi Nagatomi. Anatomic variations of the cerebral arteries and their embryology: a pictorial review. European radiology, 12:2548-256 2561, 2002.
[81] Ana Maria Dumitrescu, Claudia Florida Costea, Cristina Furnică, Mihaela Dana Turliue, Andrei Ionut Cucu, Camelia Margareta Bogdânici. Şerban Turliue, Gabriela Florenja Dumitrescu, and Anca Sava. Morphological aspects of the vasculogenesis and angiogenesis during prenatal edification of the circle of willis: review. Romanian Journal of Morphology and Embryology. 62(3)6679, 2022.
14

---

[82] JJ Van Overbeeke, B Hillen, and CA Tulleken. A comparative study of the circle of willis in fetal and adult life. the configuration of the posterior bifurcation of the posterior communicating artery. Journal of Anatomy, 176:45, 1991.
[83] Zhenhong Liu, Xingce Wang, Zhongke Wu, Yi-Cheng Zhu, and Alejandro F Frangi. Mri joint super- resolution and denoising based on conditional stochastic normalizing flow. IEEE Transactions on Artificial Intelligence, 2024.
[84] Luca Antiga, Bogdan hordahce, and Andrea Remuzzi. Computational geometry for speceitt- reconstruction and meshing of blood vessels from mr and ct angiography. JEEE transactions on medical imaging, 22(5):674 684, 2003.
[85] Scott MacDonald Black, Craig Maclean, Pauline Hall Barrientos, Konstantinos Ritos, and Asimina Kazakidi. Reconstruction and validation of arterial geometries for computational fluid dynamics using multiple temporal frames of 4d flow-mri magnitude images. Cardiovascular Engineering and Technology, 14(5):655 676, 2023.
[86] Richard Izzo, David Steinman, Simone Manini, and Luca Antiga. The vascular modeling toolkit: a python library for the analysis of tubular structures in medical images. Journal of Open Source Software, 3(25):745, 2018.
[87] Arunachalam Narayanaswamy, Saritha Dwarakapuram. Christopher S Bjomsson, Barbara Cutler, William Shain, Badrinath Roysam. Robust adaptive 3-d segmentation of vessel laminae from fluo- rescence confocal microscope images and parallel gpu implementation. IEEE transactions on medical imaging, 29)3):583-597, 2009.
[88] Michal Chlebiej, Anna Zurada, Jerzy Gielecki, Mikolaj A Pawlak, and Maciej Szkulmowski. Customizable ubbular model for n-furcating blood vessels and its application to 3d reconstruction of the cerebrovascular system. Medical & Biological Engineering & Computing, 6146)113331136, 2023.
[89] Dieuwertje Alblas, Iris N Vos, Micha M Lipplaa, Christoph Brune, Irene C van der Schaaf, Mireille RE Velthuis, Birgitta K Velthuis, Hugo Kuijf, Ynte M Ruigrok, and Jelmer M Wolterink. seee-leamingg-asee extraction circle of willis topology with anatomical priors. Scientific reports, 14(1):31630, 2024.
[90] Md Shakib Shahariar Junayed, Kazi Shahriar Sanjid, Md Tanzim Hossain, M Monir Uddin, and Sheikh Anisul Haque. Topology-aware anatomical segmentation the circle of willis: Hunet unveils the vascular network. IET Image Processing. 18(10):2745- 275222745-7722222222 2024.
[91] Chih- Yang Hsu, Mahsa Ghaffari, Ali Alaraj, Michael Flannery, Xiaohong Zhou, and Andreas Linninger. Gap-free segmentation of vascular networks with automatic image processing pipeline. Computers in biology and medicine, 82:29-31 39, 2017.
[92] Jacob R Bumgarner and Randy Nelson. Open-source analysis and visualization of segmented vasculature datasets with vesselvio. Cell reports methods, 2(4), 2022.
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A Background
A.1 Geometric Deep Learning and Continuous Function Learning
Geometric deep learning 27 is a branch of deep learning that leverages deep neural networks to model and extract intrinsic geometric features of structured data. To date, geometric deep learning has yielded impressive results in a range of fields, notably physics, biology, and computer graphics. In physiscs, GNS [63] and MESHGRAPHNET [64] investigate the application of geometric deep learning in the simulation of fluid and cloth, aiming to capture their intricate physical dynamics. In biology, DeepDock [65] and its enhanced version, caDeepDock [66], predict the binding conformation by learning the distributions based on Euclidean distance. In computer graphics, PointNet (24) and PointNet++ [55] represent major breakthroughs in point cloud processing by effectively capturing geometric features directly from raw point clouds. Most geometric deep learning methods operate on discrete data in Euclidean space, such point clouds, by constructing geometric graphs(67] [88 or defining local point neighborhoods. These approaches typically assign geometric attributes (e.g.. spatial coordinates) to the nodes of the graph and the points, enabling the extraction of meaningful geometric features.
A topological manifold [69| is special type of geometric space, formally defined topologiaal space that is locally Euclidean. Compared to modeling discrete data such as point cloud in 3D Eu- clidean space, modeling directly on continuous manifold domains can avoid the errors introduced by numerical estimation or implicit representations learned via neural networks. This approach enables models to achieve greater expressiveness, interpretability, and generalization capability. Function learning models 770 71,, with the goal of constructing high-dimensional mappings from functions to functions, preserve the analytical properties of the functions themselves, such smoothness, and has found increasing application 770 71 Inspired by the aforementioned approaches, we construct a 3D rotation and translation equivariant mapping on smooth surface manifolds represented by Ball B-Spline CuBVC(BBSC) [8). Owing to its compactness. local modifiability, smoothness, convex hull property, the BBSC provides an efficient and accurate representation of tubular geometric structures.
A.2 Circle of Willis
CoW is a ring-shaped arterial structure located at the base of the brain in birds, reptiles, and mammals 72 It supplies blood to the brain and surrounding structures by connecting the bilateral internal carotid artery (ICA) systems with the vertebrobasilar system. In cases of stenosis or occlusion of a major artery (e.g.. the ICA), the CoW provides collateral circulation to maintain perfusion in ischemic regions, serving a compensatory function. Additionally, the CoW plays a critical role in balancing and regulating intracranial arterial pressure and in protecting against acute cerebrovaseular events 773. Notably, the CoW is also a frequent site for cerebral aneurysms, which, when ruptured, can lead to stroke and potentially fatal outcomes. In its idealized form. the standard CoW consists of several key arteries: the anterior cerebral arteries (left and right), the anterior communicating artery, the distal termini of the internal carotid arteries (left and right), the PI segments of the posterior cerebral arteries (left and right), and the posterior communicating arteries (left and right) (74). Depending on clinical needs, more detailed sub-regional definitions may be adopted. However, due to factors such as genetic variation and developmental anomalies, the CoW often exhibits significant topological heterogeneity its anatomical structure. For example, incomplete CoWs may result from the absence of one or more arterial segments; developmental variants may manifest as arteries that are duplicated, abnormally narrow, or overly dilated. From a topological perspective, the CoW is generally classified into three categories: complete circles, partially closed, and open configurations. Among them, open CoWs are often associated with poor collateral capacity and are more prone to widespread ischemia or infarction in the presence of arterial narrowing, occlusion, aneurysm rupture. This high degree of topological variability poses considerable challenges for clinical diagnosis and increases the importance of automated and robust computational methods.
The segmentation and classification of the CoW are both critical and well-suited computational approaches. Accurate and efficient methods not only reduce the workload of medical professionals but also enhance diagnostic precision, playing an essential role the detection and early monitoring of CoW-related cerebrovascular diseases. To the best of our knowledge, there are currently few computational models specifically designed for CoW segmentation and classification. Although some methods developed for general vascular analysis can be adapted, their performance CoW remains
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unstable due to its inherent topological heterogeneity [59] 60 61]. Therefore, designing effective
models for CoW-related tasks requires joint analysis of both geometric and topologiaal features.
Such models 75 9 76 are crucial for capturing the structural variations and delivering reliable diagnostic support in real-world clinical scenarios.
B
Preliminary Model Architecture
B.1 DiffusionNet
DiffusionNet [46] is an effective method learn point cloud features using the principle of heat diffusion. It mainly consists of two modules: diffusion range learning and gradient feature learning.
Learning diffusion. The diffusion learning can be interpreted
filter that allows each point to
propagate information radially and the diffusion range (uo))) with initial state no. mainly depends on learnable diffusion time described by formula H4 e ug. DiffusionNet effectively discretizes and simulates the diffusion process using the implicit Euler time step method,
hu( = (M + +IL)
(16)
where M and represent mass matrix and weak Laplace matrix. To avoid the inconvenience of solving large linear system separately for each Channel,Diffusional offers spectral acceleration which leveages the precomputed basis of low frequency Laplacian eigenfunctions represent the diffusion approximately. Let + [ox] E R. V v represent the concatenated matrix of the eigenvectors corresponding to the first k smallest eigenvalues X; where [d.] are solutions to L.d; = dr The coefficients of spectral basis can be obtained by Th Mu and u can be recovered by &c, and the coefficients of after t time diffusion can calculated by formula of = c. Then diffusion can be simplified as follows,
h (u) = Tbe = [( -dot Ail
e AubyT (T)
(17)
Learning gradient feature. In additional, DiffusionNet constructs a learning scaling gradient feature for each point enhance the feature learning ability of the filter. First, the gradient operator which is assembled into a sparse matrix G. The matrix is independent of the vertex feature and can be precomputed for each shape by computing the least-squares approximation of the function values of each point and its neighbors on the tangent plane. Note that in order to conveniently represent the tangent vector in the calculation of the gradient matrix, DiffusionNet uses a complex expression. Multiply each channel feature of vertex by the gradient matrix G and stack them up to get the gradient feature w, [Gu]. Then a learned scaling gradient feature g. shown as follow:
D ΣΣ 2=1.a1
D
w. (.(() ) A AER RXD (18)
P. = -ttan((Raal (we, Aw,))) tanh
Real
where A is the learned real matrix which describe the scaling and D is the number of the feature channel. Whether to choose a real or complex matrix A mainly depends on whether the point has a normal. Because multiplying complex scalar do both the scaling and rotationtransformations The real part of the scalar represents scaling, while the imaginary part represents rotation. Generally, the direction of rotation deponds on the outward normal and orientation, so for points without normal, real matrix A is sufficient.
It is worth noting that though only low-frequency basis is used in the diffusion spectral accelera- tion part, DiffusionNet can still learn figh-freuency features through gradient learning and MLPs (Multi-Layer Perceptrons).
B.2 TranssTTanformerr
The TranssTrr (30] integrates the principles of 3D rotation and translation equivariance by embedding Tensor Field Network (TFN) (29 into the computation of the key, query, and value vectors within the Transformer architecture. This design endows the Transformer with SE{() equivariance in the 3D Euclidean space, ensuring that its outputs are consistent under rigid transformations. First, the TranssTransformer leverages spherical harmonics [37] to map 3D point coordinates into a higher- dimensional representation space. Spherical harmonics form a set of orthogonal Fourier bases defined
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on the unit sphere S². Specifically, an degree spherical harmonics Y' can map any 3D unit vector to 21 1dimeniiis vector in 2R f1, expressed (z) [X"", (.)...,, Y
(z), Y (z)).
Then, the Wigner-D matrices 77 which are unitary and irreducible, are introduced to achieve equivariance of high-dimensional vectors generated by spherical harmonics under rotations in SO(3):
(R = (T)) E SO(3), R3x3 ('( E (2(++))x((+11)
(19)
Furthermore, (lebcch-Gordan tensor product (38 provides mechanism to aggregate vectors from spaces of different dimensions while preserving equivariance. For V [(2(++))xC) and V 1)) Vil V Slyg V can be expanded as:
C1,C2
Vi m,C Σ
Wer c02c
Σ
Σ
Clam (ll.))) (2.m1))mmm)) mm Ving
(20)
1,02ml
where Q (1.m) ((1.my)(la.m) are the Clebsch-Gordan coefficients.
SE(3)-Transformer constructs set of learnable radial functions : R20 R and leveages the Clebsch- Gordan coefficients form the basis kernel W (2 (2e,+)+)) which can transform
(2c; + 1) channels input feature to (2c, + 1) channels output feature:
y E
We
s/ (1EE))
(21)
(=/o-col
where Y is the mith element in Y', Qime R (2) With the basis kernel W Ca channels input feature Ji of points v, Trans-eraneo can generate a co channels feature four as shown in Equation and (5).
C Proof of Infinite Differentiability in BBSC Mapping
Proposition 1. Let U = (P E R 33xn, Ru,, E T). where = (tz E [o. 1lltpt1 < xp+2 < < En f1 =... fp+1 = 0, t,
intpt1 1}. Then, the mapping 3 U + M, defined according to Equation ① and Equation (2), a mapping.
Proof. The p-degree BBSC manifold can be written as:
B(Prr,T)= - (t,((t) r.) = Σ (t)(t) Σ Ntpp()rrs = (tt)),
(22)
1=0
1=0
and the basis function is follows:
t- t Ni.p-1(t)
1+pp11 -
1,
it 4411, (23)
(t) = tstp ti
N 1i)),
tupp11 t++1
( otherwise.
where P EE E R,1pt1 < <lp+2 < ... < bn, = -... 4p41 0,4n .... = batr+1 = 1. We prove the smoothness of the manifold M by showing that 3 admits infinitely many continuous partial derivatives with respect to P, r, and т. is a linear combination of the control points Pt, and thus admits infinitely many continuous partial derivatives with respect to each P., i.e., VP, E R.3
03 OP,
03 OP?
8003
(t),
0.
(24)
0p
Similarly, since 3 admits infinitely many continuous partial derivatives with respect each control radius r;.
For т, to avoid the denominator being O, we do not consider the endpoints of the BBSC which means we only consider the differentiability of strictly increasing sequence tp+1. fp-:....-....... fn. Now we
need to prove that the corresponding basis function N.,p which contains 4; has continuous infinite partial derivatives. As observed in Equation (3 the basis function Ni,p is recursively defined
composite function, and its differentiation can be handled via the chain rule. Since each knot value 4 appears in the basis function only through two types of expressions, simply denoted as fi t-ti
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and f2 = 1-b (maybe the former needs to be added with minus sign, but this does not affect the 4--6 derivative calculation), it is sufficient to verify whether both types possess infinitely many continuous partial derivatives. Thus 3 admits infinitely many continuous partial derivatives with respect knot vector 7 and can be calculated as follows:
dm f1 dt
m!
m! (t b) dth f2 (-1)" de
# te
+ t,
(25)
(a (a-t,) m+
(t, (t--bu b)
By the chain rule, any higher-order derivative of Nip with respect to i, can be expressed as combination of products and sums of the derivatives of the fi and f2.
For P and the function is separable with respect to these variables. Specifically, it can be written the product of G(P) and (T) N,()):
= N(T)G(P)'
(26)
Its mixed partial derivatives then expressed
83 am 8m d d Opm dpm 0r dPm dr where m
(27)
Since the partial derivatives of G(P) and (T)) exist for all orders and are continuous, the mixed partial derivatives of ß with respect to P and are also continuous for all orders.
Similarly, arbitrary-orde continuous mixed partial derivatives with respect and can be obtained. For P and r, we have
c(P) N2(1)P"= N,(())" (P) where H(r)
(28)
which ensures that also admits arbitrary-order continuous mixed partial derivatives with respect to P and r. The same reasoning applies to r itself. Thus, it can be proven that V/ E E,, 3 is O mapping.
D Details of the CoW Dataset and Training Procedure
D.1 Per-Class Distribution of Dataset
TepCoW 2024 Class Distribution
Clinical Class Distribution
120
2000
800
80
Number
400
20
30
1.
30
20
Labe
Labal
Figure 6: Per-class distribution of TopCoW 2024 and clinical dataset.
The CoW exhibits highly heterogeneous anatomical structure, which arises not only from inter-ethnic and inter- inter-populations variability [8] but also from genetic factors embryonic development vascular remodeling (82. This inherent
environmental influences(81] and hemodynamies-drive heterogeneity leads significant class imbalance in data representations of the CoW. As illustrated in Figure
6 we present an overview of the class distribution in both the TopCoW 2024 MICCAI Challenge dataset and our internal clinical dataset. Notably, in the TopCoW dataset, category 12 severely underrepresented, with vessels labeled class 12 comprising only 5% of all samples. Even
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when training the model with uniformly stratified split between training and testing sets, the model's learning of class-12 12 vessels remains highly stochastic- e.gg.. . in extreme cases, the test set might contain only a single sample with class-12 branches. In contrast, as shown in Figure 4 our method demonstrates robust performance on the clinical dataset, even for rare classes such as and 21, achieving accurate segmentation despite their limited presence in the data. While data augmentation a common strategy to address such imbalance, we currently do not employ it. This is primarily due to the use of SE(3) (eq)--qquva architectures, which are inherently invariant to translations and rotations, rendering many conventional augmentations redundant in our context.
D.2 Preprocess of Clinical Data
Cenoiso Segement
Meah Generate
Figure 7: Clinical data preprocessing pipeline. A mesh is constructed from each MRI scan. The denoise and clean operations are performed to remove noise. Then the branches of the CoW are segmented. Surface point clouds and centerlines computed from the segmented mesh, followed by BBSC fitting.
Compared with the TopCoW2024 dataset, clinical CoW scans are more challenging due to imaging artifacts and spurious small vessel components. Consequently, series of preprocessing steps is required before BBSC fitting and shown in Figure 7 First, when severe MRI artifacts such as motion blur metal-induced distortions degrade mesh quality, denoising methods be applied (83) Vascular meshes are then reconstructed from MRI either using geometric methods 84 85 or the
VMTK package [86). The reconstructed meshes often contain noise from small vessel components and exhibit disconnected fragments. These are removed through cleaning [87 and repaired using mesh fix algorithms [8]) Branch segmentation [89 90] and ground truth labeling follow, with manual
verification and correction by domain experts to ensure accuracy. The majority of preprocessing operations can be implemented with open-source packages such as VMTK or medical CAD tools [2]. Finally, surface point clouds and centerlines are extracted from the cleaned meshes for BBSC fitting, It is worth noting that, in our pipeline, the mesh is first reconstructed and then subjected to cleaning, denoising, and segmentation. Alternatively, these preprocessing operations can also be applied directly the MRI images prior mesh reconstruction.
D.3 Training Detail
BBSC construction. For each branch of the CoW, we individually construct BBSC representation. These are then processed using DiffusionNet to obtain the three key components of the BBSC: control points, control radius, and the knot vector. For notational convenience, denote the concatenation of control points and radius as the control parameters, represented as 4D vectors. In general, the length of the control parameters either proportional to the branch's length determined by threshold-based allocation strategy. However, in our experiments, we observed that assigning fixed control parameter length of 13 for all branches significantly simplified and accelerated the BBSC construction process, while having negligible impact on downstream classification performance. When constructing BBSCs, we consistently the spline degree we assign all branches control parameter length of 13. For the clinical dataset from hospital, we adopt a length allocation scheme based on are length thresholds, assigning control parameter lengths of 13, 10, and 5, respectively. Additionally, during the BBSC fitting process, we adopt the computationally efficient and sosy-to-implemntt L2 norm metric (Equation (7). The term || - Pl] in this metric also be replaced with (P, P), yielding comparable fitting performance. The
For the TopCoW 2024 dataset,
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weighting parameters and 1) are set to 1. During training, all control parameters are padded to fixed length of 13 to ensure consistency across samples. It important note that we pad with zeros. To distinguish between padded zeros and the zero entries in the knot vector, we apply value shift of +1 to all elements in the knot vector.
BESG))-BBCTTrM Due to the local support property of the p-degree B- B-Spline, where the influence of the i-th control parameter is restricted ((.:p...iipp11) we first construct an (p 1) matrix F, where the i-th row corresponds to [ti..... di+p+1] 1] (p = 3). We then apply several MLP jointly to F and the control radius to produce (21 + 1) diddennsio vector, where | = 0, 1......... Cio (cr, 3). The direct sum of Ct vectors constitutes the rotation and translation equivariant input fir of SB33)-BBSCfrrmer defined in Section 4.3. The degree of spherical harmonics is also Cip. The architecture of our graph convolutional network (GCN) follows a hybrid of the GCN and Res-GCN design used in DeepDock, as illustrated in Figure 8
(we set H1 = 3 and H2 = 10).
Hix
GCN
H₂x Residual GCN
Figure 8: Architecture of GCN in SEC3)-BSCCOOmeGGNN
Train and test detail. We train and evaluate all models presented in this paper on a single NVIDIA 3090Ti GPU. For the BBB(3))BBCCoorrGGCC we set the number of (E(3)-BBSCforme layers 1. During training, we used the Adam optimizer with a fixed learning rate of 0.001 and a weight decay of 0.01. The number or workers is 1. The loss function is set to cross cross-entroe entropy for all models, and the number of training epochs is fixed at 200. For the TopCoW 2024 dataset, lightweight models such as JBS))BBBsrrrrrr, BBSCformerGCN, and SEL3)--BSCCormeeGGN are trained and tested using a 5-fold cross-validation protocol, and their mean and standard deviation is computed and showed in Figure 3 During training, we split the TopCow 2024 dataset into 100 samples for training and 20 for testing. Similarly, the clinical dataset is partitioned into training and test sets with a 4:1 split.
E Per-Class AUC-ROC
Label AUC-ROC Distribution
3.95
AUC-ROC
3.85
3.75
Labe
Figure 9: Per-class AUC-ROC of various models evaluated on the TopCow 2024 Circle of Willis dataset.
We further present the AUC-ROC scores of different models across various label categories on both the TopCoW 2024 dataset and a clinical dataset, aiming to assess the per-class classification performance in more detail. As shown in Figure I0 and Figure + the y-axis is restricted to the range of 0.75 to 1.0 for improved visual clarity, allowing for more effective comparison between models in terh-peerformacce regimes. BBC3)-BBSCfommeCCCC AUC-ROC scores across all anatomical label categories, clearly outperforming baseline methods such as voxel-based and point-based networks. This consistent superiority observed not only on the relatively amall-scale TopCoW 2024 dataset, but also on the large-scale real-world clinical dataset,
consistently achieves the highest
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Figure 10: Per-class AUC-ROC of various models evaluated on the Circle of Willis dataset collected from clinical imaging at hospital.
highlighting the model's strong generalization ability. The results further demonstrate the advantages of our BBSC representation and the SE(3)-equivarian architecture, which together enable precise modeling of complex geometric and topological structures while maintaining robustness across domains. Importantly, even for challenging categories with high anatomical variability, our model maintains high AUC-ROC scores, indicating its effectiveness capturing fine-grained structural differences and mitigating overfitting.
F
Case Study of Clinical Data
We conduct a case study on Circle of Willis (CoW) clinical data collected from a collaborating medical institution, focusing on samples where BEC)oBBBCCNNN makes prediction errors and which exhibit certain topological heterogeneity. As shown in Figure 1I the first row presents complete CoW structure, a configuration that only accounts for about 20% 25% of real-world cases. In the remaining samples with topological variations, most commonly observed is the absence of the bilateral posterior communicating arteries (PcoA-L and PcoA-R), R), which connect the upper and lower parts of the CoW. In the fourth row, more severe case of unilateral absence is observed.
Beyond topology, geometric heterogeneity within the same type of branch also poses challenges. For instance, the MCA-L2 branch highlighted in green in the third row is noticeably longer than its counterparts in other samples, while the ICA-RI branch the fourth row, marked in ochre, is significantly thicker than in the other cases. Such variations in both topology and geometry contribute the clinical difficulty of accurately classifying CoW branches.
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Ground Turth
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Figure 11: Clinical CoW samples with topological heterogeneity and the corresponding mispredicted structures by (BC3)-BBSCfoomeCCCC The first two columns show the complete CoW structures, while the last two columns highlight the misclassified branches.
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Reviews:
[{"review_id": "zmpCkPLGBs", "reviewer": "Reviewer_FYAa", "summary": "This 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.", "questions": "1. 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?", "limitations": "1. 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).", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "1. 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.", "was_revised": true, "final_justification": "I gave this paper a borderline accept. While it contains some minor issues—such as formatting inconsistencies and presentation flaws—I found the core ideas to be potentially valuable and worth further exploration. However, the paper falls somewhat outside my primary area of expertise, and as such, my confidence in assessing its technical depth and novelty is limited. Another reviewer recommended rejection, citing multiple mathematically inaccurate, confusing, and incorrect statements. Given these contrasting evaluations and the technical nature of the concerns, I strongly recommend that a meta-reviewer (or two) with deeper expertise in this specific area take a close look, in order to provide a fair and well-informed judgment of the paper’s correctness and overall contribution.", "rebuttal": "We sincerely appreciate your positive comments on our work and the invaluable suggestions provided. We address your insightful questions point by point as follows:\n\nQ 1: Ambiguity in topology and mechanism to handle it.\n\nA1：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.\nWe 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]. \nGraph 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.\n\nQ2: Overlook of the SE(3)-equivariant in other areas of medical imaging .\n\nA2: 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.\n\nHuman 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. \nWe 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.\n\nQ3: Whether the geometry and topology have learned clinical variables. Can the method be used as a biomarker?\n\nA3: 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.\nIn 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.\n\nQ4: Trade-offs between smoothness and accuracy.\n\nA4: 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.\n\n1. 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).\n\n2. 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.\n\n3. 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.\n\nThank 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.\n\nL1: Complexity and training stability \n\nA1: Thank you for pointing out the potential limitations regarding training stability, failure cases, and convergence behavior.\nIn 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.\n\nL2: Scalability to noisy or incomplete data\n\nA2: 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.\n\nL3: Limited discussion of biological plausibility\nA3: 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.\n\n[1].Liu Z, et al. MRI Joint Super-Resolution and Denoising based on Conditional Stochastic Normalizing Flow.\n\n[2].Girshick R. Fast r-cnn.\n\n[3].Lv Z, et al. A Parallel Cerebrovascular Segmentation Algorithm Based on Focused Multi-Gaussians Model and Heterogeneous Markov Random Field.\n\n[4].Liu X, et al. Extending Ball B-spline by B-spline.\n\n[5]. Zhao Y, et al. G2 Blending Ball B-Spline Curve by B-Spline.\n\n[6]. Wu Z, et al. Fitting Scattered Data Points with Ball B-Spline Curves using Particle Swarm Optimization.\n\n[7].Wang X, et al. Skeleton-based cerebrovascular quantitative analysis.\n\n[8].Y Liu, et al. Automated anatomical labeling of a topologically variant abdominal arterial system via probabilistic hypergraph matching.\n\n[9].Billot B, et al. SE(3)-Equivariant and Noise-Invariant 3D Rigid Motion Tracking in Brain MRI.\n\n[10].Wang J, et al. SpaER: Learning Spatio-temporal Equivariant Representations for Fetal Brain Motion Tracking.\n\n[11].Wang F, et al. MPerformer: An SE(3) Transformer-based Molecular Perceptron."}, {"review_id": "12bYvVeRr0", "reviewer": "Reviewer_AeuR", "summary": "This 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.", "questions": "Can 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.", "limitations": "The paper mentions the lack of equivariance of GCN but does not discuss other limitations of the pipeline.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\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.", "was_revised": true, "final_justification": "Overall it is a strong paper on analysis of tubular branching structures modeled as graphs. Although I do not completely understand all the authors' clarifications, I do not penalize the paper for my lack of understanding. My major concern about using topological representations (at the end) remain but those are general, well-known concerns in the TDA community about lack of invertability of TDA features. I had originally given a strong rating to the paper originally and I maintain that rating.", "rebuttal": "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.\n\nW1: Not much topological discussion\nA1: We sincerely thank you for your suggestions regarding our topology-aware design. \nOur 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.\n\nIn 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.\n\nW2: The significance of SE(3)-equivariant to performance is unclear.\nA2: 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.\n\nW3: Is the input data relatively clean?\nA3: 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.\n\nWe sincerely appreciate your interest in our work. We address your questions as follows.\n\nQ1：Can you provide some examples of the input data?\n\nA1: 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):\n\n-0.9574  0.5936  0.6443  0.9277\n\n-0.8946  0.5826  0.6481  0.9218\n\n-0.7000  0.5475  0.6357  0.8960\n\n-0.3348  0.1458  0.5663  0.9499\n\n-0.2276 -0.0279  0.5510  1.0042\n\n 0.0665 -0.3906  0.4303  0.9124\n\n 0.3661 -0.5725  0.1682  0.9839\n\n 0.5042 -0.5007 -0.0663  0.9944\n\n 0.6324 -0.2403 -0.3195  1.0272\n\n 0.6850 -0.0188 -0.4251  0.9941\n\n 0.7426  0.2168 -0.5108  1.2511\n\n 0.7992  0.3759 -0.5876  2.1610\n\n 0.8091  0.4154 -0.6635  0.8286\n\nThe corresponding knot vector is:\n[0.0, 0.0, 0.0, 0.0, 0.0357, 0.0646, 0.1030, 0.1692, 0.2370,\n 0.2833, 0.3230, 0.3530, 0.4215, 1.0, 1.0, 1.0, 1.0]\n\nQ2: Comparison with shape graph representations.\n \nA2: 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.\n\nQ3: Discussion around the cost function and data matching term\n\nA3: 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:\n\nDuring 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.\n\n[1].Liu Z, et al. MRI Joint Super-Resolution and Denoising based on Conditional Stochastic Normalizing Flow.\n\n[2].Lv Z, et al. A Parallel Cerebrovascular Segmentation Algorithm Based on Focused Multi-Gaussians Model and Heterogeneous Markov Random Field.\n\n[3].Liu X, et al. Extending Ball B-spline by B-spline.\n\n[4].Zhao Y, et al. G2 Blending Ball B-Spline Curve by B-Spline.\n\n[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."}, {"review_id": "ZFu6MBigqv", "reviewer": "Reviewer_G1qr", "summary": "Medical 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.", "questions": "See the above section.", "limitations": "Yes.", "rating": 2, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 1, "significance": 1, "originality": 2, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "While the reviewer believes there to be some novelty, as also recognized by the other reviewers, the quality of the current manuscript is of such nature that the reviewer has a hard time trusting that all concerns will be addressed in the final version. \nTherefore, it would still be my recommendation to reject the paper and for the authors to submit a polished version of the manuscript to another conference to give it the exposure the work deserves.", "rebuttal": "Thank you so much for your thorough review and providing us with invaluable, constructive comments to greatly improve the quality of our manuscript.\n\nIn 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.\n\nRegarding the mathematical formulation:\n\nQ1: Smoothness of mixed partial derivatives.\n\nA1: 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 $P$. \n\nQ2:  Non-standard usage.\n\nA2:  We would like to clarify that $\\beta$ is infinitely differentiable with respect to $r$, and all of its derivatives are continuous.\n\nQ3: Smoothness of BBSC.\n\nA3: 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.\n\nQ4: Bijectivity. \n\nA4: 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. \n\nQ5: How would the GCN preserve equivariant information?\n\nA5: 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.\nTo 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. \n\nRegarding your concerns about the experimental results:\n\nQ1: Confusion of the CoW size.\n\nA1: 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.\n\nQ2: Why was cross-validation not performed on the clinical dataset?\n\nA2: 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: \n\nAccuracy：$96.11 \\pm 1.01$; AUC-ROC: $99.38\\pm0.19$; Precision: $96.11\\pm0.93$; Recall: $96.08\\pm0.97$; F1 score: $96.02\\pm1.05$\n\n\nQ3: Description of the CoW dataset is unclear.\n\nA3: 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.\n\nQ4: The number of clinical data between the main text and the appendix is inconsistent.\n\nA4: 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.\nAnatomical 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.\n\nQ5: Baseline methods in related work are not compared.\n\nA5: 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.\n\nQ6: Regarding NeurIPS checklist: \n\nA6: For Q7, Table 1 reports the error bars and Figure 6 in Appendix D.1 visualizes the statistical significance of the experiments.\n\nThe following are the references (due to space constraints, only the titles are provided).\n\n[1].Liu Z, et al.MRI Joint Super-Resolution and Denoising based on Conditional Stochastic Normalizing Flow.\n\n[2]. Girshick R. Fast r-cnn.\n\n[3].Lv Z, et al.A Parallel Cerebrovascular Segmentation Algorithm Based on Focused Multi-Gaussians Model and Heterogeneous Markov Random Field.\n\n[4].Liu X, et al. Extending Ball B-spline by B-spline.\n\n[5]. Zhao Y, et al. G2 Blending Ball B-Spline Curve by B-Spline.\n\n[6].Wang X, et al. Skeleton-based cerebrovascular quantitative analysis.\n\n[7].Y Liu, et al. Automated anatomical labeling of a topologically variant abdominal arterial system via probabilistic hypergraph matching."}, {"review_id": "4L5PKgooZQ", "reviewer": "Reviewer_C8vC", "summary": "This 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.", "questions": "How 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?", "limitations": "The 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.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\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", "was_revised": true, "final_justification": "The paper introduces a new perspective in using BBSC manifold to model tubular structures. This is a novel and interesting idea. There is certain limitation in experimental settings that they are mainly arisen from the inherent challenges in collecting data.", "rebuttal": "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:\n\nW1: Limited Novelty\n\nQ1: 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.\n\nWe emphasize that our work presents a new, effective, and efficient paradigm for modeling tubular structures. We outline the key novelties as follows:\n\n1. 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.\n\n2. 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.\n\n3. 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.\n\nW2: Narrow Scope and Insufficient Evaluation\n\nA2: 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:\n\nThe 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.\nWe 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.\n\nW3: Lack of Medical Relevance Testing\n\nThank you so much for the invaluable insights. Our response are as follows:\n\n1. 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.\n\n2. 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.\n\n3. 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.\n\nWe respond to your questions as follows:\n\nQ1: How is the BBSC fitting performed?\n\nA1: 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].\n\nQ2: No real-world biomedical datasets included.\n\nA2: 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.\n\nQ3: Can the method be adapted to cases with bifurcations or looped topologies?\n\nA3: 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.\n\nQ4: No camparison with MLP and GCN.\n\nA4: 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.\n\nThank you for these invaluable constructive suggestions. We will respond to your listed limitations and incorporate the limitations you pointed out into our revised manuscript.\n\nL1：Lack of end-to-end learning.\n\nA1: 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.\n\nL2: Dependence on clean, tube-like structure assumptions.\n\nA2：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.\n\nL3: No indication of computational cost or efficiency.\n\nA3: 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.\n\nL4：Lack of applicability to non-tubular data.\n\nA4: 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.\n\nL5: No empirical evidence or discussion of the ethical implications of automated anatomical modeling.\n\nA5：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.\n\n[1].Liu Z, et al.MRI Joint Super-Resolution and Denoising based on Conditional Stochastic Normalizing Flow.\n\n[2].Lv Z, et al.A Parallel Cerebrovascular Segmentation Algorithm Based on Focused Multi-Gaussians Model and Heterogeneous Markov Random Field.\n\n[3].Liu X, et al. Extending Ball B-spline by B-spline.\n\n[4].Zhao Y, et al. G2 Blending Ball B-Spline Curve by B-Spline.\n\n[5].Wang X, et al. Skeleton-based cerebrovascular quantitative analysis.\n\n[6]. Wu Z, et al. Fitting Scattered Data Points with Ball B-Spline Curves using Particle Swarm Optimization."}]

Author rebuttal:

Meta-review: This paper introduces a novel framework for modeling tubular branching structures using BBSC manifolds. While one reviewer raised concerns about presentation quality and mathematical clarity, the majority view the contribution as interesting, technically sound, and potentially impactful. With novelty and relevance to the community, the overall balance of reviews supports acceptance. 

I would strongly encourage the authors to incorporate the new results, as well as clarifications, especially the concerns of reviewer G1qr, into the revised version.
Decision: Accept (poster)