Manifold-Aware Persistent Homology Representation for Point Cloud Registration with Topology Preservation

Point clouds sampled from real-world objects often exhibit complex non-convex geometry, in which global structure such as connectivity, holes and part adjacency is essential for shape understanding, yet topological analysis based on Euclidean distances measures proximity in ambient space rather than intrinsic connectivity and can create shortcuts across narrow gaps or concave boundaries. We introduce a manifold-aware topological representation that computes persistent homology from geodesic distances to provide faithful global shape cues, and condition a registration network on it: TopoReg routes the case-level descriptor into a standard coarse-to-fine backbone at a single deep level, so that the gain comes from the representation rather than from added capacity. We further prove that any stable descriptor built on the Euclidean filtration changes by at most an amount proportional to the gap width when a narrow gap is bridged, whereas the graph geodesic metric separates the two sides by an unbounded amount. Experiments on the large-scale lung CT dataset Lung250M-4B and the KITTI autonomous driving dataset show state-of-the-art performance and robust generalization.