用环面拟合点云，实现任意分辨率的快速符号距离计算。
Points as Tori: Fast Pointwise Signed Distance for Point Clouds
- 通过局部拟合环面构建闭式符号距离函数，支持快速点查询。
- 无需全局优化或空间离散化，可并行处理，速度显著提升。
- 适合直接在点云上进行偏移、布尔运算和球面追踪可视化。
本文提出一种快速计算点云符号距离的方法，可对任意空间位置进行点级查询。输入为带法向量的点云，输出为解析参数化表达，支持对近似隐含曲面的任意点进行符号距离查询，同时完成重建与距离计算。核心思想是通过局部拟合环面实现形状重建，因环面具有闭式符号距离函数。环面参数通过预训练网络以前馈方式输出每点的曲率与偏移量。该方法不依赖昂贵的全局优化或空间离散化，且易于并行。其理论基础统一了符号距离与经典的环绕数、泊松表面重建方法。我们将其应用于摄影测量、网格、3D高斯和神经隐式表示生成的点云。结果表明，点云可直接用于各类应用：如执行偏移、形态学与布尔操作，并通过球面追踪直接可视化偏移曲面。
原文摘要 · Abstract (English)
We describe a method for computing signed distance to point clouds that allows fast pointwise evaluation at arbitrary spatial resolution. As input, our method takes a point cloud with normals; as output, it provides an analytical parameterization that allows queries of signed distance to the approximate underlying surface at arbitrary points - simultaneously providing reconstruction and distance. Our key idea is to reconstruct shapes by locally fitting point clouds with tori, which have closed-form signed distance functions. Tori are fitted in a feed-forward manner, using a pre-trained network to output per-point curvature and shift parameters. Importantly, our method does not require costly global optimization or spatial discretization, and is easily parallelizable. Underlying our method is a new theory that unifies signed distance with the classic reconstruction methods of winding numbers and Poisson surface reconstruction. We use our method to compute signed distance to point clouds arising from photogrammetry, meshes, 3D Gaussians, and neural implicits. Our method allows point clouds to be used directly in applications, without explicit surface reconstruction: as examples, we take offsets of point clouds, apply morphological and Boolean operations, and directly visualize offset surfaces using sphere tracing.
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