Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applications

TL;DR

This paper presents new, efficient boundary estimators from point clouds with rigorous error guarantees, and demonstrates their application to PDE boundary-value problems and image data analysis.

Contribution

It introduces novel boundary estimators with proven accuracy and applies them to solve PDEs on point clouds, advancing computational methods in geometric and PDE analysis.

Findings

Estimators outperform existing methods in accuracy.

Error bounds are rigorously established for estimators.

Successful application to PDEs and image data demonstrates practical utility.

Abstract

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.