Smooth surface interpolation to scattered data using interpolatory subdivision algorithms

Abstract

In this paper, a smooth interpolatory subdivision algorithm for the generation of interpolatory surfaces (GC¹) over arbitrary triangulations is constructed and its convergence properties over nonuniform triangulations studied. An immediate application of this algorithm to surface interpolation to scattered data in $\text{R}{}^{\mathrm{n}}$, n ≥ 3 is also studied. For uniform data, this method is a generalization of the analyses for univariate subdivision algorithms, and for nonuniform data, an extraordinary point analysis is proposed and a local subdivision matrix analysis presented. It is proved that the subdivision algorithm produces smooth surfaces over arbitrary networks provided the shape parameters of the algorithm are kept within an appropriate range. Some error estimates for both uniform and nonuniform triangulations are also investigated. Finally, three graphical examples of surface interpolations over nonuniform data are given to show the smoothing interpolating process of the algorithm.