Boundary Estimation from Point Clouds: Algorithms, Guarantees and\n Applications

2021/11/04 by Jeff Calder, Sangmin Park, Calder, Jeff +3 · 1 citation
Mathematics · #62G20 #65D99 #65N12 #65N15 #65N75 #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2111.03217

openalex publication_date 2021/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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