Re-interpreting Thin-Plate Splines as Minimum Change Surfaces
by
G.N. Newsam
Defence Science & Technology Organisation

By definition, a thin-plate spline is the surface with minimum mean square second derivative energy that interpolates a given collection of points. Although the definition has no obvious claims to universality, thin-plate splines have proven to be surprisingly good interpolants of many naturally occuring data sets in one, two and three dimensions. Moreover, even when the standard interpolant fails, simple extensions of the basic concept often suffice to produce satisfactory surfaces. In particular, for the important case of interpolating point height measurements of natural terrain where the standard spline surfaces are unnaturally smooth, Hutchinson has shown that replacing the standard penalty by one that depends only on the square of the second derivative in the gradient direction produces realistic results. This paper seeks to explain some of this good performance through an alternative interpretation of thin-plate splines and their generalisations as surfaces governed by evolution equations that interpolate the data while undergoing minimal change.

More specifically, if the available data are assumed to be point samples at a given instant in time of a diffusion process, then we shall show that the standard thin-plate spline can also be interpreted as being the most slowly changing surface (as measured by its mean square rate of change), and therefore the most likely surface, that is consistent with the data. Moreover, generalising the nature of the diffusion process, e.g. to those governed by a p-Laplacian for some value of p, provides a natural way to construct interpolants to data associated with these processes. In particular, the \infty-Laplacian has been previously proposed as an idealised model for certain erosion processes, and Hutchinson's interpolant can be re-cast as the minimum mean square change surface under this process that passes through the given terrain heights. Moreover at the other extreme, the 1-Laplacian has been proposed as an ideal smoothing operator for imagery that preserves important edge structure: the re-interpretation proposed here provides a natural way of constructing the smoothest image consistent with a noisy picture.

Finally most diffusion processes arise naturally in relation to minimisation of some energy functional of the first derivatives of a surface, yet splines constructed by direct minimisation of these functionals subject to interpolating the given data points usually fail to give realistic results. The paper will close with some speculations as to the reasons behind this.

Date received: December 15, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caie-07.