Natural neighbor interpolation is the most general and robust method of interpolation available to date. It produces a conservative, artifice-free, result by finding weighted averages, at each interpolation point, of the functional values associated with that subset of data which are natural neighbors of each interpolation point. The resulting function is continuous everywhere within the convex hull of the data, and has a continuous slope everywhere except at the data themselves. For the bivariate case, this surface mimics a taut rubber sheet which is stretched to meet the data, and this is analogous to the higher dimensional results. The interpolation weights are the natural neighbor local coordinates that address the interpolation point relative to its natural neighbor data, and these coordinates are ratios of the intersection contents of Voronoi partitions; they are always positive and sum to one. So computing the contents (area, volume, hyper-volume) of irregular convex regions within the data cloud is
the fundamental computational operation required to find these natural neighbor coordinates, and a straightforward
geometrical algorithm, Natural neighbor interpolation in Cartesian space is gaining recognition as a dependable method, especially
for sparse and erratic data, and it can also be applied to data in spherical space such as angles, directions,
compositions, mixtures, or any such 'closed' data. If you are interested in the details of this algorithm, it is
more fully explained in A Dave Watson, 17 May 2002 |