Discrete box splines and refinement algorithms
This paper gives a general knotline refinement algorithm for functions which are linear combinations of translates of box splines. The Lane-Riesenfeld algorithm is obtained as a special case. The derivation is based on discrete splines. Section 2 introduces the box spline as follows. Given integers \(n\geq s\geq 1\) and vectors \(X=(x^ 1,...,x^ n)\) with \(x^ j\in\mathbb R^ s\), \(j=1,...,n\), the box spline \(M_ x=M(.| X): \mathbb R^ s\to \mathbb R\) is defined as a distribution given by the rule \[ \int_{\mathbb R^ s}M_ x(x)f(x)\,dx=\int_{I^ n}f(X_{\nu})\,d\nu_ 1\,d\nu_ 2...\,d\nu_ n \] for all \(f\in C_ 0^{\infty}(\mathbb R^ s)\), where the right-hand integration is over the unit cube \(I^ n=[0,1]^ n\) in \(\mathbb R^ n\) and \(X=\nu_ 1x^ 1+...+\nu_ nx^ n\). Some properties of the box spline are also discussed in section 2. An algorithm for knot-line refinement is derived in section 3. The algorithm for computing \(d^ r_ j=d^ n_ j\) is as follows. 1. Fill the \(d\) array with \(m^ s\)-copies for each \(P_ i\). 2. For \(r=s+1,...,n\) in each of directions \(z^{s+1},...,z^ n\) take an average of \(m\) \(d_ j^{r-1}\)'s to form \(d^ r_ j\). Some further properties of discrete box spline are considered in section 4. Some examples are given in section 5 and remarks in section 6. The last section gives a proof of control mesh convergence.
scientific article; zbMATH DE number 3905723