The role of inner summaries in the fast evaluation of thin-plate splines

Abstract

The driving force behind fast evaluation of thin-plate splines is the fact that a sum $\sum_{j=n_1}^{n_2}\lambda_j \phi(x-\xi_j)$, where $\phi(x)=\Vert{x}\Vert^2\;{\rm{log}}\; \Vert x\Vert$, can be efficiently and accurately approximated by a truncated Laurent-like series (called an outer summary) when the data sites $\{{\xi_j}\}_{j=n_1}^{n_2}$ are clustered in a disk and when the evaluation point x lies well outside this disk. We present a means (called an inner summary) of approximating this sum when the evaluation point x lies inside the disk. The benefit of having an inner summary available (and of an improved error estimate for the outer summary), is that one can safely use uniform subdivision of clusters in the pre-processing phase without concern that an unfortunate distribution of data sites will lead to an unreasonably large number of clusters. A complete description and cost analysis of a hierarchical method for fast evaluation is then presented, where, thanks to uniform subdivision of clusters, the pre-processing is formulated in a way which is suitable for implementation in a high level computing language like Octave or Matlab.