发表机构
Nanyang Technological University; Nanjing University of Aeronautics and Astronautics; Chinese Academy of Sciences; City University of Hong Kong; Texas A&M University（南洋理工大学; 南京航空航天大学; 中国科学院; 香港城市大学; 德克萨斯A&M大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出投影法向场（PNF），一种无朝向的凸优化框架，通过秩一投影器编码双向法向并松弛为半正定矩阵，结合局部拟合与正则化估计一致法向，进而构造光滑UDF，实验验证了其鲁棒性和精度。
AI 中文摘要
从原始点云构造无符号距离场（UDF）的光滑逼近具有挑战性，因为输入既不提供表面连通性，也不提供一致朝向的法向。直接学习标量UDF的方法还必须处理其在零水平集上的不可微性以及远离样本处的弱监督问题，这可能导致优化不稳定和空间伪影。我们引入了投影法向场（PNFs），一种无朝向的表示和凸优化框架，仅从点位置估计双向法向。每个法向轴由一个秩一投影器编码，该投影器对法向反转不变。我们将硬投影器的非凸集松弛到其凸包：具有单位迹的对称半正定矩阵。每个软张量定义了一个局部二次距离模型，并保留了候选法向轴的相对权重。我们通过结合局部切平面拟合、软PCA锚定和固定邻接图上的重叠正则化来估计一致的PNF。当锚定权重为正时，目标函数是强凸的，并且具有唯一的全局最小化器。主特征向量提供双向法向，而相应的特征间隙提供谱置信度指标。我们使用这些指标来选择和加权用于热扩散的方向源，随后进行泊松积分以构造正则化的UDF逼近。通过将局部几何估计与标量场构造分离，PNF避免了直接拟合不可微的UDF。实验表明，该方法对邻域大小的敏感性降低，在噪声和离群值下具有竞争力的重建性能，并且在非流形连接附近提高了精度。项目页面可在以下https URL获取。
英文摘要
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/
