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Issue ESAIM: Proc.
Volume 20, 2007
RFMAO 05 - Rencontres Franco-Marocaines en Approximation et Optimisation 2005
Page(s) 16 - 28
DOI http://dx.doi.org/10.1051/proc:072002
Published online 13 October 2007

ESAIM: Proc., 2007, Vol. 20, pp. 16-28
DOI: 10.1051/proc:072002

Subdivision Scheme of Quartic Bivariate Splines on a Four-Directional Mesh

El Bachir Ameur1, Domingo Barrera Rosillo2 and Driss Sbibih3

1  Université Moulay Ismail, Faculté des sciences et Techniques, Département d'Informatique, 52000 Errachdia, Maroc
2  Departamento de Matemática Aplicada, Faculdad de Ciencias, Universidad de Granada, Campus Universitario de Fuentenueva s/n, 18071, Granada, Spain
3  Université Mohammed I, Ecole Supérieure de Technologie, Laboratoire MATSI, Oujda, Maroc

ameurelbachir@yahoo.fr
dbarrera@ugr.es
sbibih@yahoo.fr

(Published online: 13 October 2007)

Abstract
In this paper we give a new definition of minimally and quasi-minimally supported $\mathcal{C}^{2}$ quartic bivariate B-splines associated with the four-directional mesh of the plane, introduced in [7,19], which is convenient to show that theses B-splines satisfy the refinement equation and we determine the associated matrix mask, we prove that the family of these B-splines is stable and the associated subdivision scheme converges. These results can be extended to various cases in the spline space of class $\mathcal{C}^{3k+2}$ and degree 4k+4, but in these cases the supports of the masks are larger.


Résumé
Dans cet article nous donnons une nouvelle définition des B-splines quartiques de classe $\mathcal{C}^{2}$ à support minimal et quasi-minimal sur le réseau quadridirectionnel du plan, introduites dans [7,19]. Nous utilisons cette définition pour montrer que ces B-splines vérifient une équation de raffinement et déterminer les matrices du filtre associé. Nous montrons que la famille de ces B-splines est stable et que le schéma de subdivision associé converge. Ces résultats peuvent être généralisés aux espaces de fonctions splines de classe $\mathcal{C}^{3k+2}$ et de degré 4k+4, mais dans ce cas les supports des filtres associés deviennent plus grands.


Mathematics Subject Classification. 41A05, 41A15, 65D05, 65D07

Key words: B-splines, Box-splines, Four-directional mesh, Refinement equation, Near-best quasi-interpolants


© EDP Sciences, ESAIM 2007


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