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

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.