A sampling theorem for MLS surfaces

Peer Timo Bremer, John C Hart

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish (US)
Title of host publicationPoint-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings
Pages47-54
Number of pages8
StatePublished - Dec 1 2005
EventEurographics/IEEE VGTC Symposium on Point-Based Graphics, 2005 - Stony Brook, NY, United States
Duration: Jun 20 2005Jun 21 2005

Publication series

NamePoint-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings

Other

OtherEurographics/IEEE VGTC Symposium on Point-Based Graphics, 2005
CountryUnited States
CityStony Brook, NY
Period6/20/056/21/05

Fingerprint

Sampling
Surface reconstruction

ASJC Scopus subject areas

  • Engineering(all)

Cite this

Bremer, P. T., & Hart, J. C. (2005). A sampling theorem for MLS surfaces. In 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.

Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings. 2005. p. 47-54 1500317 (Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Bremer, PT & Hart, JC 2005, A sampling theorem for MLS surfaces. in 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.
Bremer PT, Hart JC. A sampling theorem for MLS surfaces. In Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings. 2005. p. 47-54. 1500317. (Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings).
Bremer, Peer Timo ; Hart, John C. / A sampling theorem for MLS surfaces. Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings. 2005. pp. 47-54 (Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings).
@inproceedings{a628f8af266f4241b28c4cdb3e70b29e,
title = "A sampling theorem for MLS surfaces",
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.",
author = "Bremer, {Peer Timo} and Hart, {John C}",
year = "2005",
month = "12",
day = "1",
language = "English (US)",
isbn = "3905673207",
series = "Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings",
pages = "47--54",
booktitle = "Point-Based Graphics, 2005 - Eurographics/IEEE VGTC Symposium Proceedings",

}

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

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

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 -