通过显式建模中轴线，提升点云表面距离函数的全局精度。
Medial Axis Aware Learning of Signed Distance Functions
- 基于高阶变分法，显式考虑梯度间断集（即中轴线）
- 在近场与全局范围内均实现高精度距离函数重建
- 适合需要精确几何表示的3D建模与逆向工程场景
我们提出一种新颖的变分方法，用于计算给定点云的高精度全局有符号距离函数（SDF）。为此，通过高阶变分公式显式考虑SDF梯度的跳跃集，该跳跃集与曲面的中轴线一致，并在远离此不连续集的方向上强制梯度呈线性增长。同时施加Eikonal方程和SDF零水平集作为约束条件。为使该变分问题可计算，采用Ambrosio-Tortorelli型相场近似，其关联的相场函数隐式描述中轴线。方法在无向点云表示的曲面上实现，使用神经网络分别近似SDF与相场函数。实验表明，该方法在近场与全局范围内均具有高精度。定量与定性对比显示其优于现有方法。
原文摘要 · Abstract (English)
We propose a novel variational method to compute a highly accurate global signed distance function (SDF) to a given point cloud. To this end, the jump set of the gradient of the SDF, which coincides with the medial axis of the surface, is explicitly taken into account through a higher-order variational formulation that enforces linear growth along the gradient direction away from this discontinuity set. The eikonal equation and the zero-level set of the SDF are enforced as constraints. To make this variational problem computationally tractable, a phase field approximation of Ambrosio-Tortorelli type is employed. The associated phase field function implicitly describes the medial axis. The method is implemented for surfaces represented by unoriented point clouds using neural network approximations of both the SDF and the phase field. Experiments demonstrate the method's accuracy both in the near field and globally. Quantitative and qualitative comparisons with other approaches show the advantages of the proposed method.
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