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

In this paper we give a new definition of minimally and quasi-minimally supported C 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 C 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 C 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 C et de degré 4k + 4, mais dans ce cas les supports des filtres associés deviennent plus grands. Introduction Refinable function vectors and vector subdivision schemes, as two of the most important and extensively studied fundamental objects in the literature of wavelet analysis, are useful in many applications such as signal processing and computer aided geometric design ( [4,5,9,10,13,16,22]). In [9,10] the authors give the subdivision scheme associated with the function vectors of two linear and cubic B-splines. In this paper we are interested in the function vector of three C2 quartic B-splines on the four-directional mesh. We begin with some notations and definitions used throughout this paper. Let τ be the uniform triangulation of the plane, whose set of vertices is Z, and whose edges are parallel to the four directions e1 = (1, 0), e2 = (0, 1), e3 = (1, 1) and e4 = (1,−1). This type of triangulation is called a four-directional mesh. For any integers r and d, let Pd be the space of bivariate polynomials of total degree at most d, and Sr d(τ) := { s ∈ Cr(R2) : s|T ∈ Pd for all T ∈ τ } 1 Université Moulay Ismail, Faculté des sciences et Techniques, Département d’Informatique, 52000 Errachdia, Maroc;email: [email protected] 2 Departamento de Matemática Aplicada, Faculdad de Ciencias, Universidad de Granada, Campus Universitario de Fuentenueva s/n, 18071, Granada, Spain; e-mail: [email protected] 3 Université Mohammed I, Ecole Suprieure de Technologie, Laboratoire MATSI, Oujda, Maroc; e-mail: [email protected] c © EDP Sciences, SMAI 2007 Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc:072002 ESAIM: PROCEEDINGS 17 be the space of bivariate piecewise polynomial functions of class Cr on the plane and whose restrictions to each triangular cell of τ are in Pd. Let T be a triangle of τ and λ = (λ1, λ2, λ3) be the barycentric coordinates of a point M of R relative to T . Each polynomial p of Pd(T ) has a unique representation in the Bernstein-Bézier form: p(M) = ∑ μ∈4d b(μ) B μ(λ) where 4d = { μ = (μ1, μ2, μ3) ∈ Z+ : |μ| = μ1 + μ2 + μ3 = d } and B μ(λ) = d! μ! λ = d! μ1!μ2!μ3! λ1 1 λ μ2 2 λ μ3 3 . The family of the ( d+2 2 ) polynomials B μ, μ ∈ 4d, forms a basis for the space Pd(T ). The coefficients {b(μ), μ ∈ 4d}, are called the B-net of p on the triangle T . When r = 0 and d = 1, the linear bivariate splines space on the four-directional mesh τ , S0 1 (τ), is generated by two minimally supported linear B-splines φ1 and φ2, whose B-nets and supports are given in Figure 1.