Improving continuity of Voronoi-based interpolation over Delaunay spheres

AbstractThere are two types of the discontinuity of the original Voronoi-based interpolants: one appears on the data sites and the other on the Delaunay spheres. Some techniques are known for reducing the first type of the discontinuity, but not for the second type. This is mainly because the second type of the discontinuity comes from the coordinate systems used for the interpolants. This paper proposes a sequence of new coordinate systems, called the kth-order standard coordinates, for all nonnegative integers k, and shows that the interpolant generated by the kth-order standard coordinates have Ck continuity on the Delaunay spheres. The previously known Voronoi-based interpolants coincide with the cases k=0 and k=1. Hence, the standard coordinate systems constructed in this paper can reduce the second type of the discontinuity as much as we want. In addition, this paper derives a formula for the gradient of the standard coordinates