Numerical Procedures for Surface Fitting of Scattered Data by Radial Functions

In many applications one encounters the problem of approximating surfaces from data given on a set of scattered points in a two-dimensional domain. The global interpolation methods with Duchon's “thin plate splines” and Hardy's multiquadrics are considered to be of high quality; however, their application is limited, due to computational difficulties, to $ \sim 150$ data points. In this work we develop some efficient iterative schemes for computing global approximation surfaces interpolating a given smooth data. The suggested iterative procedures can, in principle, handle any number of data points, according to computer capacity. These procedures are extensions of a previous work by Dyn and Levin on iterative methods for computing thin-plate spline interpolants for data given on a square grid. Here the procedures are improved significantly and generalized to the case of data given in a general configuration.The major theme of this work is the development of an iterative scheme for the construction of a smooth surface, presented by global basis functions, which approximates only the smooth components of a set of scattered noisy data. The novelty in the suggested method is in the construction of an iterative procedure for low-pass filtering based on detailed spectral properties of a preconditioned matrix. The general concepts of this approach can also be used in designing iterative computation procedures for many other problems.The interpolation and smoothing procedures are tested, and the theoretical results are verified, by many numerical experiments.