Data dependent triangulations for piecewise linear interpolation

Given a set of data points in R² and corresponding data values, it is clear that the quality of a piecewise linear interpolation over triangles depends on the specific triangulation of the data points. While conventional triangulation methods depend only on the distribution of the data points in R² in this paper we suggest that the triangulation should depend on the data values as well. Several data dependent criteria for defining the triangulation are discussed and efficient algorithms for computing these triangulations are presented. It is shown for a variety of test cases that data dependent triangulations can improve significantly the quality of approximation and that long and thin triangles, which are traditionally avoided, are sometimes very suitable.