Truncated Laurent expansions for the fast evaluation of thin plate splines

Abstract

Thin plate splines are highly useful for the approximation of functions of two variables, partly because they provide the interpolant to scattered function values that minimizes a 2-norm of second derivatives. On the other hand, they have the severe disadvantage that the explicit calculation of a thin plate spline approximation requires a log function to be evaluatedm times, wherem is the number of “r ²logr ²” terms that occur. Therefore we consider a recent technique that saves much work whenm is large by forming sets of terms, and then the total contribution to the thin plate spline from the terms of each set is estimated by a single truncated Laurent expansion. In order to apply this technique, one has to pick the sets, one has to generate the coefficients of the expansions, and one has to decide which expansions give enough accuracy when the value of the spline is required at a general point of ℓ². Our answers to these questions are different from those that are given elsewhere, as we prefer to refine sets of terms recursively by splitting them into two rather than four subsets. Some theoretical properties and several numerical results of our method are presented. They show that the work to calculate all the Laurent coefficients is usuallyO(m logm), and then onlyO(logm) operations are needed to estimate the value of the thin plate spline at a typical point of ℓ². Thus substantial gains over direct methods are achieved form⩾200.