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.

翻译：我们提出了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了SDF梯度的跳跃集（即曲面中轴），该公式在该不连续集沿着梯度方向之外强制线性增长。将程函方程和SDF的零水平集作为约束条件实施。为使得该变分问题在计算上可行，采用了Ambrosio-Tortorelli类型的相场近似，对应的相场函数隐式描述了中轴。该方法针对无方向点云表示的曲面实现，利用了神经网络对SDF和相场两者的近似。实验证明了该方法在近场和全局范围内的精度。与其他方法的定量和定性比较显示了所提方法的优势。