Smooth unsigned distance fields built from point clouds with convex optimization
Projective Normal Fields: A Convex Optimization Method for Constructing Smooth UDFs
Computer Vision and Pattern Recognition
Summary
Creating smooth surfaces from raw 3D point clouds is hard because the points don’t show how the surface connects or if normals point consistently. The authors propose Projective Normal Fields, a new way to estimate directions of surfaces without worrying about normal orientation. Using this approach, they build a smooth distance field that better represents the shape and reduces errors, especially near tricky spots like intersections. This method also makes the math easier to solve because it uses a kind of optimization that always finds the best answer.
What this means in practice
- •For 3d scanning engineers: Improve surface reconstruction from unstructured point clouds for creating smoother 3D models in scanning workflows.
- •For computer graphics developers: Generate more accurate distance fields to support rendering and simulation tasks requiring smooth surface approximations.
Authors
Jiayi Kong, Chen Zong, Fei Hou, Junhui Hou, Wenping Wang, Ying He
Abstract
Constructing a smooth approximation of an unsigned distance field (UDF) from a raw point cloud is challenging because the input provides neither surface connectivity nor consistently oriented normals. Methods that directly learn a scalar UDF must also handle its non-differentiability on the zero level set and weak supervision away from the samples, which can lead to unstable optimization and spatial artifacts. We introduce Projective Normal Fields (PNFs), an orientation-free representation and convex optimization framework for estimating bidirectional normals from point positions alone. Each normal axis is encoded by a rank-one projector, which is invariant to normal reversal. We relax the non-convex set of hard projectors to its convex hull: the symmetric positive-semidefinite matrices with unit trace. Each soft tensor defines a local quadratic distance model and retains the relative weights of candidate normal axes. We estimate a coherent PNF by combining local tangent-plane fitting, soft-PCA anchoring, and overlap regularization on a fixed neighborhood graph. With positive anchoring weights, the objective is strongly convex and admits a unique global minimizer. Principal eigenvectors provide bidirectional normals, while the corresponding eigengaps provide spectral confidence indicators. We use these indicators to select and weight directional sources for heat diffusion, followed by Poisson integration to construct a regularized UDF approximation. By separating local geometry estimation from scalar-field construction, PNF avoids directly fitting the non-differentiable UDF. Experiments demonstrate reduced sensitivity to neighborhood size, competitive reconstruction under noise and outliers, and improved accuracy near non-manifold junctions. The project page is available at https://anonymous17777367.github.io/PNF-page/