Natural neighbor (or Sibson) interpolation is a method of spatial interpolation, developed by Robin Sibson.[1] The method is based on Voronoi tessellation of a discrete set of spatial points. This has advantages over simpler methods of interpolation, such as nearest-neighbor interpolation, in that it provides a smoother approximation to the underlying "true" function.
The basic equation is:
where is the estimate at , are the weights and are the known data at . The weights, , are calculated by finding how much of each of the surrounding areas is "stolen" when inserting into the tessellation.
where A(x) is the volume of the new cell centered in x, and A(xi) is the volume of the intersection between the new cell centered in x and the old cell centered in xi.
where l(xi) is the measure of the interface between the cells linked to x and xi in the Voronoi diagram (length in 2D, surface in 3D) and d(xi), the distance between x and xi.
Natural neighbor interpolation has also been implemented in a discrete form. This discrete form has been demonstrated to be computationally more efficient in at least some circumstances.[4] A form of discrete natural neighbor interpolation has also been developed that gives a measure of interpolation uncertainty.[5]
There are several useful properties of natural neighbor interpolation:[5]