RODR: Riemannian Orthogonal Decoupling Representation for Feature-Preserving Continuous Surface Recovery

Continuous surface recovery from discrete, noisy point clouds is fundamental to 3D representation learning. Despite empirical progress in flow matching, score-based diffusion, and coordinate regression, existing paradigms suffer from a structural objective-geometry conflict: isotropic Euclidean losses couple extrinsic surface-normal attraction with intrinsic tangential spacing, precipitating contractive collapse and quadratic off-manifold drift. In this work, we propose Riemannian Orthogonal Decoupling Representation (RODR), a lightweight, architecture-agnostic geometric optimization framework. RODR projects spatial vector updates strictly onto local normal bundles and tangent bundles, mathematically eliminating first-order cross-subspace gradient interference. Coupled with a multi-scale geometric planarity gate, RODR dynamically throttles tangential velocity near structural discontinuities, locking in sharp CAD boundaries without over-smoothing. Crucially, continuous flow optimization under RODR enables neural velocity fields to internalize the underlying Riemannian metric directly into network parameters, avoiding the finite-sample covariance noise of test-time discrete geometric projections. Extensive evaluations across standard benchmarks, high-density CAD models (up to 200K points), deterministic regressors, and stochastic differential equations demonstrate that RODR cuts surface manifold thickness (Local Manifold Fitting Residual, LMFR) by up to 38.2%, delivering metrology-grade downstream mesh reconstructions while accelerating inference throughput with a compact 0.53M backbone.