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Laboratoire d'InfoRmatique en Image et Systèmes d'information
UMR 5205 CNRS / INSA de Lyon / Université Claude Bernard Lyon 1 / Université Lumière Lyon 2 / École Centrale de Lyon
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International journal with reviewing committee

L-system specification of knot-insertion rules for non-uniform B-spline subdivision [PDF]
Vincent Nivoliers [INRIA Lorraine - LORIA] , Cédric Gérot [GIPSA-lab] , Victor Ostromoukhov [LIRIS] , Neil Stewart [Université de Montréal]
2/2012
Computer Aided Geometric Design 29(2):150-161, Elsevier B.V., ISSN 01678396.

Résumé

Subdivision schemes are based on a hierarchy of knot grids in parameter space. A univariate grid hierarchy is regular if all knots are equidistant on each level, and irregular otherwise. We use L-systems to design a wide class of systematically described irregular grid hierarchies. Furthermore, we give sufficient conditions on the L-system which guarantee that the subdivision scheme, based on the non-uniform B-spline of degree d defined on the initial knot grid, is uniformly convergent. If n is the number of symbols in the alphabet of the L-system, this subdivision scheme is defined with a finite set of masks (at most n^(d+1)) which does not depend on the subdivision step. We provide an implementation of such schemes which is available as a worksheet for Sage software.

Abstract

Subdivision schemes are based on a hierarchy of knot grids in parameter space. A univariate grid hierarchy is regular if all knots are equidistant on each level, and irregular otherwise. We use L-systems to design a wide class of systematically described irregular grid hierarchies. Furthermore, we give sufficient conditions on the L-system which guarantee that the subdivision scheme, based on the non-uniform B-spline of degree d defined on the initial knot grid, is uniformly convergent. If n is the number of symbols in the alphabet of the L-system, this subdivision scheme is defined with a finite set of masks (at most n^(d+1)) which does not depend on the subdivision step. We provide an implementation of such schemes which is available as a worksheet for Sage software.

BibTex

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@Article{Liris-5454,
  title         = {{L-system specification of knot-insertion rules for 
    non-uniform B-spline subdivision }},
  author        = {Vincent {Nivoliers} and Cédric {Gérot} and Victor 
    {Ostromoukhov} and Neil {Stewart}},
  year          = {2012},
  month         = feb,
  journal       = {Computer Aided Geometric Design},
  volume        = {29}, 
  number        = {2}, 
  pages         = {150--161}, 
  publisher     = {Elsevier B.V.}, 
  DOI           = {http://dx.doi.org/10.1016/j.cagd.2011.11.004}, 
  issn          = {01678396}, 
  language      = {en},
  url           = {http://liris.cnrs.fr/publis/?id=5454}
}