A consistent and yet practically accurate definition of curvature onto polyhedral meshes remains an open problem. We propose a new framework to define curvature measures, based on the Corrected Normal Current, which generalizes the normal cycle: it uncouples the positional information of the polyhedral mesh from its geometric normal vector field, and the user can freely choose the corrected normal vector field at vertices for curvature computations. A smooth surface is then built in the Grassmannian ℝ3 × 𝕊2 by simply interpolating the given normal vector field. Curvature measures are then computed using the usual Lipschitz–Killing forms, and we provide closed‐form formulas per triangle. We prove a stability result with respect to perturbations of positions and normals. Our approach provides a natural scale‐space for all curvature estimations, where the scale is given by the radius of the measuring ball. We show on experiments how this method outperforms state‐of‐the‐art methods on clean and noisy data, and even achieves pointwise convergence on difficult polyhedral meshes like digital surfaces. The framework is also well suited to curvature computations using normal map information.

中文翻译：

---

多边形表面的插值校正曲率测量

在多面体网格上一致且实际准确的曲率定义仍然是一个悬而未决的问题。我们提出了一个新的框架来定义曲率度量，基于修正法向电流，它概括了法向循环：它将多面体网格的位置信息与其几何法向矢量场解耦，用户可以自由选择修正后的法向矢量场在顶点进行曲率计算。然后通过简单地内插给定的法向量场，在格拉斯曼 ℝ3 × 𝕊2 中构建光滑表面。然后使用通常的 Lipschitz-Killing 形式计算曲率度量，我们为每个三角形提供封闭形式的公式。我们证明了关于位置和法线扰动的稳定性结果。我们的方法为所有曲率估计提供了一个自然的尺度空间，其中刻度由测量球的半径给出。我们在实验中展示了这种方法如何在干净和嘈杂的数据上优于最先进的方法，甚至在数字表面等困难的多面体网格上实现逐点收敛。该框架也非常适合使用法线贴图信息进行曲率计算。