Surface Reconstruction from Unoriented Points via Green's 3rd Identity

Surface reconstruction from unoriented point clouds is a fundamental problem in geometry processing. While recent winding-number-based approaches, such as Parametric Gauss Reconstruction (PGR) and Anisotropic Gauss Reconstruction (AGR), provide elegant implicit formulations, they often suffer from underdetermined or ill-conditioned linear systems. In this work, we analyze the theoretical root of this instability: the winding number formula is oblivious to a broad family of tangential perturbations of the normal field, which introduces a non-trivial nullspace in the resulting integral equations. To address this, we present G3R, a simple yet effective method based on Green’s third identity for harmonic functions. This approach eliminates the tangential nullspace with a basis of harmonic functions, resulting in a well-conditioned linear system which can be solved directly without additional regularization. Furthermore, we apply a Gaussian mollification of the fundamental solution, which handles near-field singularities robustly. We adopt a matrix-free implementation and accelerate it with a hierarchical scheme to scale our method to large point clouds. Experiments demonstrate that our approach achieves highly competitive performance in both normal orientation accuracy and surface reconstruction quality. Our code is publicly available at https://github.com/ne0-wu/G3R.