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符号距离函数（SDF）。为此，我们通过高阶变分公式明确考虑SDF梯度的跳跃集（该集合与曲面的中轴重合），该公式在远离此间断集的方向上沿梯度方向施加线性增长约束。SDF的程函方程和零水平集作为约束条件强制执行。为使该变分问题在计算上可行，采用了Ambrosio-Tortorelli型相场近似，其关联的相场函数隐式描述中轴。该方法针对无向点云表示的曲面实现，通过神经网络同时近似SDF和相场。实验表明该方法在近场和全局均具有高精度。与其他方法的定量和定性比较展示了所提方法的优越性。