Characterization of bivariate hierarchical quartic box splines on a three-directional grid

Abstract

We consider the adaptive refinement of bivariate quartic C 2-smooth box-spline spaces on the three-directional (type-I) grid G. The polynomial segments of these box splines belong to a certain subspace of the space of quartic polynomials, which will be called the space of special quartics. Given a finite sequence (G ℓ) ℓ=0,...,N of dyadically refined grids, we obtain a hierarchical grid by selecting cells from each level such that their closure covers the entire domain Ω, which is a bounded subset of R 2. A suitable selection procedure allows to define a basis spanning a hierarchical box spline space. As our main result, we derive a characterization of this space. More precisely, under certain mild assumptions on hierarchical grid, the hierarchical spline space is shown to contain all C 2-smooth functions whose restrictions to the cells of the hierarchical grid are special quartic polynomials.

Completeness characterization of Type-I box splines

“…The results we present in this article generalize our previous work [39] on quartic box splines. Our results apply to type-I box splines of any polynomial degree with no restriction on the symmetry of their support.…”

“…Theorem 28 is a consequence of Corollary 26, and therefore of Lemma 22. This result generalizes the completeness property of the space of translates of the quartic box spline B 2 proved in [39,Theorem 26].…”

“…We now prove that for any n ∈ Z 3 , the subspace V n | ⊆ P n is independent of the choice of the triangle ∈ T . The proof of this result is a generalization of [39,Proposition 27], where we consider the case 2 = (2, 2, 2) and the box spline B 2 .…”

“…In Section 5 we construct the hierarchical type-I meshes and the corresponding hierarchical box spline spaces. This construction follows the approach presented in [31] and [39]. We conclude the paper with some final remarks in Section 6.…”

“…It has also been adapted to Powell-Sabin splines [37], Zwart-Powell elements and B-spline-type basis functions for cubic splines on regular grids [43]. In an earlier article, we constructed a hierarchical basis for quartic C 2 -continuous box splines [39]. Quartic hierarchical box splines spaces have also been studied and used for surface fitting applications by Kang, Chen and Deng in [23] and [24].…”