| Sibson's interpolant uses Voronoi diagrams in the plane to interpolate a smooth surface from given scattered data points. This natural neighbor interpolation method is particularly interesting because it adapts easily to non-uniform and highly anisotropic data, and it has several desirable properties such as linear precision, locality and C1 continuous. For this reason, Sibson's interpolant has been widely used in computational geometry, visualization, geology, geophysics modeling, and other different fields. In this dissertation, a new surface approximation method, an updated least squares Sibson's interpolation with greedy data selection strategy using Sibson's interpolant, is presented. And as part of its applications this method is applied on data representation, image reconstruction and surface approximation. Furthermore, this dissertation introduces double Sibson's interpolant by adding and weighting more neighbors around the interpolated data point. Its fundamental mathematical properties as well as its applications to scattered data interpolation, and the results compared with Sibson's interpolant, are shown in this dissertation. In addition, the basic properties and the generation of natural neighbor interpolation with different natural neighbor coordinates are investigated. Finally, the flexibility and veritable properties of natural neighbor interpolation are integrated into a framework that can extrapolate data set beyond its convex hull. |
