HyperGCN-Vessel: Hybrid Geometric Representation Learning for Topology-Preserving Vascular Segmentation

Accurate vascular segmentation requires not only high voxel-wise accuracy but also the preservation of the underlying vessel topology. Although recent deep learning models achieve impressive Dice scores, they frequently produce fragmented predictions on thin and distal vessels, limiting their utility in downstream clinical analysis. We attribute this limitation to the mismatch between conventional Euclidean feature representations and the hierarchical organization of vascular trees. To address this challenge, we propose **HyperGCN-Vessel**, a hybrid Euclidean–hyperbolic framework that combines efficient local feature extraction with geometry-aware representation learning in hyperbolic space. Specifically, the proposed framework introduces three key components: (1) an *early hybrid geometric encoder* that progressively embeds vascular representations into the Poincaré ball to better capture hierarchical structures; (2) *Topo-Gate*, an uncertainty-aware differentiable topology-preserving module that recovers disconnected vessel branches without explicit skeletonization; and (3) *HyperConv*, an efficient linear-complexity tangent-space convolution operator that enables scalable and numerically stable hyperbolic feature learning for high-resolution 3D volumes. Extensive experiments on the TubeTK dataset demonstrate that HyperGCN-Vessel substantially improves vascular connectivity while maintaining competitive segmentation accuracy, reducing the ${\beta }_0$ Betti number error by 65% compared with nnU-Net and achieving inference efficiency comparable to conventional Euclidean architectures.