Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applications
This work addresses the challenge of boundary estimation in computational geometry and PDE solving for applications like image processing, offering incremental improvements with theoretical guarantees.
The paper tackles the problem of identifying domain boundaries from point cloud data by introducing new estimators for boundary normals, distances, and boundary strip tests, which are more accurate than existing methods. It also applies these estimators to solve boundary-value problems for PDEs on point clouds, providing rigorous error estimates and numerical validation.
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.