Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applications

Jeff Calder; Sangmin Park; Dejan Slepčev

doi:10.1007/s10915-022-01894-9

Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applications

Jeff Calder, Sangmin Park, Dejan Slepčev

School of Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

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.

Original language	English (US)
Article number	56
Journal	Journal of Scientific Computing
Volume	92
Issue number	2
DOIs	https://doi.org/10.1007/s10915-022-01894-9
State	Published - Aug 2022

Bibliographical note

Funding Information:
JC was supported by NSF grant DMS 1944925, the Alfred P. Sloan Foundation, and a McKnight Presidential Fellowship. SP and DS were supported by NSF grant DMS 1814991.

Funding Information:
The authors would like to thank Eddie Aamari for valuable comments, and the anonymous referees for their helpful suggestions. The authors are also grateful to CNA of CMU, IMA of Univ. of Minnesota, and Simons Institute at UC Berkeley for hospitality.

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

Boundary detection
Distance to boundary
Meshfree methods
PDE on point clouds

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10.1007/s10915-022-01894-9

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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.",

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AU - Slepčev, Dejan

N1 - Funding Information: JC was supported by NSF grant DMS 1944925, the Alfred P. Sloan Foundation, and a McKnight Presidential Fellowship. SP and DS were supported by NSF grant DMS 1814991. Funding Information: The authors would like to thank Eddie Aamari for valuable comments, and the anonymous referees for their helpful suggestions. The authors are also grateful to CNA of CMU, IMA of Univ. of Minnesota, and Simons Institute at UC Berkeley for hospitality. Publisher Copyright: © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

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N2 - 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.

AB - 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.

KW - Boundary detection

KW - Distance to boundary

KW - Meshfree methods

KW - PDE on point clouds

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Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applications

Abstract

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