Multiresolution representations of high-dimensional scattered data is an outstanding problem in the field of scientific visualization. This paper introduces a data hierarchy of Voronoi diagrams as a versatile solution. We implemented two programs to demonstrate this, the first of which uses a constant function to approximate the data within each Voronoi tile and the second program uses the Sibson interpolant within each tile [5].

Multiresolution representation of high-dimensional scattered data is a fundamental problem in scientific visualization. This paper introduces a data hierachy of Voronoi diagrams as a versatile solution. Given an arbitrary set of points in the plane, our goal is the construction of an approximation hierarhy using the Voronoi diagram as the essential building block. We have implemented two Voronoi diagram-based algorithms to demonstrate their usefulness for hierarchical scattered data approximation. The first algorithm uses a constant function to approximate the data within each Voronoi cell, and the second algorithm uses the Sibson interpolant [14].

Betti numbers can be used as a means for feature detections to aid in the exploration of complex large-scale data sets. We present a fast algorithm for the calculation of Betti numbers for triangulated isosurfaces, along with examples of their use. Once any isosurface is extracted from a data set, calculating Betti numbers only requires time and space proportional to the isosurfaces, not the data set. Because the overhead of obtaining Betti numbers is small, our algorithm can be used with large data.

We present a new algorithm for terrain modeling based on Voronoi diagrams and Sibson's inerpolant. Starting with a set of scattered sites in the plane with associated function values defining a height field, our algorithm constructs a top-down hierarchical representation. Sibson's interpolant is used to approximate the underlying height field based on the associated function values of selected subsets of the 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. For every level of resolution our approxiamations are C1 continuous, except at the selected sites, where only C[?] continuity is satisfied. Considering n sites, the expected time complexity of our algorithm is O (n log n) . In addition to a hierarchy based on convex cells and an importance ranking for sites.