On Scattered Data Representations Using Bivariate Splines

The objective of this chapter is to present a study of scattered data representation using bivariate splines. First, we open a discussion with emphasis on the optimal order of approximation. When the polynomial degree is allowed to be sufficiently large as compared to the order of smoothness, it is shown that the spline elements can be used to represent scattered data with the optimal order of approximation over arbitrary triangulations. In real applications, where the polynomial degree is often required to be lower, it is necessary to find a so-called optimal triangulation so that the spline space can achieve the optimal approximation order. We present an algorithm to transform an arbitrary triangulation of the sample points into an optimal triangulation for representation of the scattered data using C ¹ quartic splines. Then, we consider the possibilities of finding optimal triangulations for even lower degree spline spaces such as C ¹ cubic and C ¹ quadratic spaces. Some interpolation schemes and a stable local basis construction are also presented. Finally, we mention some recent results on representing scattered data using other spline elements, such as splines on spheres and natural splines.