ABSTRACT
Thin plate spline interpolation to functions of two variables is useful in many applications, because there are few restrictions on the positions of the data points. Further, some smoothness properties are achieved naturally, because the interpolant minimizes a second derivative norm subject to the interpolation conditions. On the other hand, full matrices occur, and the number of data points, n say, may be very large. Therefore we approximate each Lagrange function by a Lagrange function of interpolation to a small subset of the data. Thus each approximation usually has far fewer than n thin plate spline terms, and the approximations provide an initial estimate of the required interpolant which can be improved by iterative refinement. Details are given of an iteration and of the use of updating techniques for the efficient calculation of the coefficients of the approximate Lagrange functions. However, we have not yet developed an automatic way of generating the subsets that have been mentioned. Some numerical results are presented too, which illustrate the number of iterations and the amount of computation of the algorithm. They suggest that interpolation to tens of thousands of scattered data points in two dimensions may soon become a routine calculation.
