Trajectory Flow: Geometry-Constrained Surface Reconstruction via Unsigned Distance Fields.

Neural implicit methods reconstruct surfaces from point clouds by learning a continuous function, usually signed or unsigned distances. For unsigned distance field (UDF) re construction, many self-supervised methods typically estimate distance through local, gradient-guided displacement regression from query points to the surface. Because these corrections are local, they can become unstable near the zero level set and accumulate errors under sparse or non-uniform sampling. We address this limitation by reformulating UDF learning as a geometrically constrained trajectory-evolution problem, shifting from static local regression to global path-consistent modeling. Specifically, we construct a bidirectional linear flow that promotes shortest-path trajectories between surface samples and query points. Under this formulation, distance prediction is interpreted as continuous state evolution along deterministic trajectories, inherently preserving geometric consistency and mitigating local error accumulation. This perspective connects UDF estimation to neural ODEs, where geometry is recovered through continuous dynamics, with the learned transport following structured trajectories induced by an implicit geometric field. To address sparse and non-uniform sampling, we further introduce a flow-guided iterative densification strategy that progressively upgrades shortest-path modeling from discrete point-to-point approximations to accurate point-to-surface projections, enabling implicit geometric completion. Extensive experiments on synthetic benchmarks and real-world scans demonstrate that our method achieves state-of-the-art performance in both reconstruction accuracy and robustness. Our source code is available at https://github.com/yccszbd/Trajectory-Flow.