Projective Normal Fields: A Convex Optimization Method for Constructing Smooth UDFs

2026-09-28

AI Summary

This study addresses the optimization instability encountered when constructing unsigned distance fields (UDFs) from raw point clouds, which arises from absent normal orientations and the non-differentiability of the zero-level set. To overcome these challenges, this work proposes a projected normal field representation coupled with a convex optimization framework. Specifically, it introduces rank-one projectors invariant to normal reversal, relaxing non-convex hard projections into convex hulls to ensure strong convexity and global optimality. By integrating soft PCA anchoring, overlap regularization, heat diffusion, and Poisson integration with spectral confidence weighting, the method achieves robust field construction. The proposed approach significantly reduces neighborhood sensitivity, maintains high-fidelity reconstruction under noise and outliers, and effectively improves geometric accuracy at non-manifold junctions.

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/

Problem

• unsigned distance field
• point cloud
• smooth approximation
• non-differentiability
• unoriented normals

Innovation

• Unsigned Distance Field
• Convex Optimization
• Projective Normal Fields
• Point Cloud Reconstruction
• Poisson Integration