Fast Evaluation of Splines Using Poisson Formula

We consider the problem of data interpolation or smoothing using a thin-plate spline approach. The thin-plate spline formulation involves the linear combination of logarithmic kernels. For problems with large data sets, current methods for evaluating the splines are too slow. Recently, Beatson and Newsam 1] have suggested using a form of multi-pole type technique for the rapid evaluation of thin-plate splines. The contribution of this paper is to demonstrate an alternative approach motivated by the work of Anderson 2]. This approach replaces the multi-pole expansion with the evaluation of an expansion along a closed contour surrounding the centres of the kernels.