Data-Dependent Triangulation in the Plane with Adaptive Knot Placement

In many applications one is concerned with the approximation of functions from a finite set of scattered data sites with associated function values. We describe a scheme for constructing a hierarchy of triangulations that approximates a given data set at varying levels of resolution. Intermediate triangulations can be associated with a particular level of a hierarchy by considering their approximation errors. We present a data-dependent triangulation scheme using a Sobolev norm to measure error instead of the more commonly used root-mean-square (RMS) error. Triangles are split by selecting points in a triangle, or its neighbors, that are in areas of potential discontinuites or areas of hight gradients. We call such points “significant points.”