Creating Surfaces from Scattered Data Using Radial Basis Functions

This paper gives an introduction to certain techniques for the construction of geometric objects from scattered data. Special emphasis is put on interpolation methods using compactly supported radial basis functions. x1. Introduction We assume a sample of multivariate scattered data to be given as a set X = solid to these data will be the range of a smooth function s : IR d ! IR D with s(x k) = y k ; 1 k N: (1) Likewise, an approximating curve, surface, or solid will make the diierences s(x j) y j small, for instance in the discrete L 2 sense, i.e. N X k=1 ks(x k) y k k 2 2 should be small. Curves, surfaces, and solids will only diier by their appropriate value of d = 1; 2, or 3. We use the term (geometric) objects to stand for curves, surfaces, or solids. Note that we assume objects to be deened via explicit representations. For implicit representations we refer the reader to Section 4. The main goal of this contribution is to describe a exible class of objects that allows construction from scattered data in a very general way. Special emphasis is put on practical aspects, while theoretical background information will be contained in a forthcoming survey, as far as it is not contained in earlier