Discrete Sibson interpolation

Natural- neighbor interpolation methods, such as Sibson's method, are well- known schemes for multivariate data fitting and reconstruction. Despite its many desirable properties, Sibson's method is computationally expensive and difficult to implement, especially when applied to higher- dimensional data. The main reason for both problems is the method's implementation based on a Voronoi diagram of all data points. We describe a discrete approach to evaluating Sibson's interpolant on a regular grid, based solely on finding nearest neighbors and rendering and blending d- dimensional spheres. Our approach does not require us to construct an explicit Voronoi diagram, is easily implemented using commodity three- dimensional graphics hardware, leads to a significant speed increase compared to traditional approaches, and generalizes easily to higher dimensions. For large scattered data sets, we achieve two- dimensional ( 2D) interpolation at interactive rates and 3D interpolation ( 3D) with computation times of a few seconds.