Terrain Modeling using Voronoi Hierarchies

We present a new algorithm for terrain modeling based on Voronoi diagrams and Sibson’s interpolant. Starting with a set of scattered data sites in the plane with associated function values defining a height field, our algorithm constructs a top-down hierarchy of smooth approximations. We use the convex hull of given sites as domain for our hierarchical representation. Sibson’s interpolant is used to approximate the underlying height field based on associated function values of selected subsets of the data sites. Therefore, our algorithm constructs a hierarchy of Voronoi diagrams for nested subsets of the given sites. The quality of approximations obtained with our method compares favorably to results obtained from other multiresolution algorithms like wavelet transforms. Our Approximations for every level of resolution are continuous, except at the selected sites, where only -continuity is satisfied. The expected time complexity of our algorithm is for sites when applying simple acceleration methods. In addition to a hierarchy of smooth approximations, our method provides a cluster hierarchy based on convex cells and an importance ranking for sites.