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Discrete Sibson Interpolation

  • Author(s): Park, Sung
  • Linsen, Lars
  • Kreylos, Oliver
  • Owens, John D.
  • Hamann, Bernd
  • et al.
Abstract

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 three-dimensional interpolation(3D) with computation times of a few seconds.

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