Bases of T-meshes and the refinement of hierarchical B-splines

Abstract

Abstract. In this paper we consider spaces of bivariate splines of bi-degree (m, n) with maximal order of smoothness over domains associated to a two-dimensional grid. We define admissible classes of domains for which suitable combinatorial technique allows us to obtain the dimension of such spline spaces and the number of tensor-product B-splines acting effectively on these domains. Following the strategy introduced recently by Giannelli and Jüttler, these results enable us to prove that under certain assumptions about the configuration of a hierarchical T-mesh the hierarchical B-splines form a basis of bivariate splines of bi-degree (m, n) with maximal order of smoothness over this hierarchical T-mesh. In addition, we derive a sufficient condition about the configuration of a hierarchical T-mesh that ensures a weighted partition of unity property for hierarchical B-splines with only positive weights.

Structural Information Reconstruction of Distorted Underwater Images Using Image Registration

“…Besides, the coarse-to-fine method, which determines the resolution of controlled mesh, is employed to achieve the optimal balance between the model flexibility and computational cost [29,30]. The optimal solution of transformation can be calculated by the formulas given in the previous publication [21].…”

Bivariate hierarchical Hermite spline quasi-interpolation

“…. , y N 2 +d 2 } ⊃ π 2 of the two enlarged intervals I e 1 ⊃ I 1 and I e 2 ⊃ I 2 , and we require f ∈ C (1,1) (Ω e ) , where now Ω e := I e 1 × I e 2 . By using the notation introduced in the previous section, the two intermediate approximations Q 1 (f ) and Q 2 (f ), can be written as follows,…”

“…The following lemma gives an upper bound for the absolute value of each local linear functional λ J (f ) and it will be later useful to develop the study of the approximation power of the hierarchical extension of Q(f ), Lemma 1. There exist four positive constants κ r,s , r , s = 0, 1 depending on d but not depending on h , such that for any function f ∈ C (1,1) (Ω e ), and for each multi-index J ∈ Γ d,h it is…”

“…Remark 3. The statement of Theorem 1 first of all says that Q H mantains on a cell c ∈ G k the maximal approximation order of Q : C (1,1) (Ω) → V k . Furthermore, in order to guarantee that Q H in S H provides the same approximation quality of Q in V M −1 , it also suggests to use larger cells where f (d 1 +1,0) and f (0,d 2 +1) are almost zero and finer elsewhere.…”

Bicubic hierarchical B-splines: Dimensions, completeness, and bases

“…Because hierarchical B-splines are generated directly from the T-meshes rather than the spline spaces, the completeness can not be guaranteed generally. Therefore, several articles [3,4,15,22,23] were published to discuss this question.…”