We investigate identifying the boundary of a domain from sample points in the domain. We introduce new estimators for the normal vector to the boundary, distance of a point to the boundary, and a test for whether a point lies within a boundary strip. The estimators can be efficiently computed and are more accurate than the ones present in the literature. We provide rigorous error estimates for the estimators. Furthermore we use the detected boundary points to solve boundary-value problems for PDE on point clouds. We prove error estimates for the Laplace and eikonal equations on point clouds. Finally we provide a range of numerical experiments illustrating the performance of our boundary estimators, applications to PDE on point clouds, and tests on image data sets.

翻译：我们从域内的采样点调查域的边界。 我们引入了正常矢量到边界的新测算器, 点点到边界的距离, 并测试某一点是否位于边界地带内。 测算器可以有效计算, 并且比文献中的测算器更准确。 我们为测算器提供严格的误差估计。 此外, 我们使用所探测到的边界点来解决点云上的 PDE 的边界值问题。 我们证明点云上的 Laplace 和 eikonal 等式的误差估计。 最后, 我们提供一系列数字实验, 说明边界测算器的性能、 对点云的 PDE 应用, 以及图像数据集的测试 。