Scattered data interpolation by box splines

Zuowei Shen and Shayne Waldron


Given scattered data in $\Rs$, interpolation from a dilated box spline space $S_M(2^k\cdot)$ is always possible for a fine enough scaling. For example, for the Lagrange function of a point $\gth$ one could take any shifted dilate $M(2^k\cdot-j)$ which is nonzero at $\gth$ and zero at the other interpolation points. However, the resulting interpolant, though smooth (and local), will consist of a set of ``bumps'', and so by any reasonable measure provides a poor representation of the shape of the underlying function. On the other hand, it is possible to choose a space of interpolants which contains some $M(2^k\cdot-j)$ of arbitrarily large support. But the resulting methods are increasingly less local, and in general still require some splines with a much higher level of dilation. Here we provide a multilevel method which constructs a space of interpolants by taking as many splines as possible from a given dilation level, then as many from the next (higher) dilation level, and so forth. The choice at each level is made using the suggestion of \cite{W99}, which is based on the Riesz representation theorem. This requires an inner product on the ground space $S_M$, and the higher levels $S_M(2^k\cdot)\ominus S_M(2^{k-1}\cdot)$, $k=1,2,\ldots$. The inner products used here involve the box spline coefficients, and prewavelet coefficients of \cite{RS92}, respectively, and are norm equivalent to $\norm{\cdot}_{L_2(\Rs)}$. These lead to a scheme which is easily implemented, and numerically stable. Previously, box spline interpolants have been considered only for data on a regular grid.

Keywords: box spline, multilevel approximation, adaptive methods

Math Review Classification: Primary 65D05, 65D07; Secondary 42C40

Length: 23 pages

Last Updated: 10 October 2007