### Abstract

Recently, point set surfaces have been the focus of a large number of research efforts. Several different methods have been proposed to define surfaces from points and have been used in a variety of applications. However, so far little is know about the mathematical properties of the resulting surface. A central assumption for most algorithms is that the surface construction is well defined within a neighborhood of the samples. However, it is not clear that given an irregular sampling of a surface this is the case. The fundamental problem is that point based methods often use a weighted least squares fit of a plane to approximate a surface normal. If this minimization problem is ill-defined so is the surface construction. In this paper, we provide a proof that given reasonable sampling conditions the normal approximations are well defined within a neighborhood of the samples. Similar to methods in surface reconstruction, our sampling conditions are based on the local feature size and thus allow the sampling density to vary according to geometric complexity.

Original language | English (US) |
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Title of host publication | Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings |

Pages | 47-54 |

Number of pages | 8 |

State | Published - Dec 1 2005 |

Event | Eurographics/IEEE VGTC Symposium on Point-Based Graphics, 2005 - Stony Brook, NY, United States Duration: Jun 20 2005 → Jun 21 2005 |

### Publication series

Name | Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings |
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### Other

Other | Eurographics/IEEE VGTC Symposium on Point-Based Graphics, 2005 |
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Country | United States |

City | Stony Brook, NY |

Period | 6/20/05 → 6/21/05 |

### Fingerprint

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings*(pp. 47-54). [1500317] (Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings).

**A sampling theorem for MLS surfaces.** / Bremer, Peer Timo; Hart, John C.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings.*, 1500317, Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings, pp. 47-54, Eurographics/IEEE VGTC Symposium on Point-Based Graphics, 2005, Stony Brook, NY, United States, 6/20/05.

}

TY - GEN

T1 - A sampling theorem for MLS surfaces

AU - Bremer, Peer Timo

AU - Hart, John C

PY - 2005/12/1

Y1 - 2005/12/1

N2 - Recently, point set surfaces have been the focus of a large number of research efforts. Several different methods have been proposed to define surfaces from points and have been used in a variety of applications. However, so far little is know about the mathematical properties of the resulting surface. A central assumption for most algorithms is that the surface construction is well defined within a neighborhood of the samples. However, it is not clear that given an irregular sampling of a surface this is the case. The fundamental problem is that point based methods often use a weighted least squares fit of a plane to approximate a surface normal. If this minimization problem is ill-defined so is the surface construction. In this paper, we provide a proof that given reasonable sampling conditions the normal approximations are well defined within a neighborhood of the samples. Similar to methods in surface reconstruction, our sampling conditions are based on the local feature size and thus allow the sampling density to vary according to geometric complexity.

AB - Recently, point set surfaces have been the focus of a large number of research efforts. Several different methods have been proposed to define surfaces from points and have been used in a variety of applications. However, so far little is know about the mathematical properties of the resulting surface. A central assumption for most algorithms is that the surface construction is well defined within a neighborhood of the samples. However, it is not clear that given an irregular sampling of a surface this is the case. The fundamental problem is that point based methods often use a weighted least squares fit of a plane to approximate a surface normal. If this minimization problem is ill-defined so is the surface construction. In this paper, we provide a proof that given reasonable sampling conditions the normal approximations are well defined within a neighborhood of the samples. Similar to methods in surface reconstruction, our sampling conditions are based on the local feature size and thus allow the sampling density to vary according to geometric complexity.

UR - http://www.scopus.com/inward/record.url?scp=33745243786&partnerID=8YFLogxK

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M3 - Conference contribution

AN - SCOPUS:33745243786

SN - 3905673207

SN - 9783905673203

T3 - Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings

SP - 47

EP - 54

BT - Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings

ER -