Leaps and Bounds: An Improved Point Cloud Winding Number Formulation for Fast Normal Estimation and Surface Reconstruction

Recent methods for point cloud surface normal estimation predominantly use the generalized winding number field induced by the normals. Optimizing the field towards satisfying desired properties, such as the input points being on the surface defined by the field, provides a principled way to obtain globally consistent surface normals. However, we show that the existing winding number formulation for point clouds is a poor approximation near the input surface points, diverging as the query point approaches a surface point. This is problematic for methods that rely on the accuracy and stability of this approximation, requiring heuristics to compensate. Instead, we derive a more accurate approximation that is properly bounded and converges to the correct value. We then examine two distinct approaches that optimize for globally consistent normals using point cloud winding numbers. We show how the original unbounded formulation influences key design choices in both methods and demonstrate that substituting our formulation yields substantive improvements with respect to normal estimation and surface reconstruction accuracy.