Natural neighbor interpolation

Natural neighbor (or Sibson) interpolation is a method of spatial interpolation, developed by Robin Sibson. 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:
 * $$G(x)=\sum^n_{i=1}{w_i(x)f(x_i)}$$

where $$G(x)$$ is the estimate at $$x$$, $$w_i$$ are the weights and $$f(x_i)$$ are the known data at $$(x_i)$$. The weights, $$w_i$$, are calculated by finding how much of each of the surrounding areas is "stolen" when inserting $$x$$ into the tessellation.


 * Sibson weights
 * $$w_i(\mathbf{x})=\frac{A(\mathbf{x}_i)}{A(\mathbf{x})}$$

where $A(x)$ is the volume of the new cell centered in $x$, and $A(x_{i})$ is the volume of the intersection between the new cell centered in $x$ and the old cell centered in $x_{i}$.


 * Laplace weights
 * $$w_i(\mathbf{x})=\frac{\frac{l(\mathbf{x}_i)}{d(\mathbf{x}_i)}}{\sum_{k=1}^n \frac{l(\mathbf{x}_k)}{d(\mathbf{x}_k)}}$$

where $l(x_{i})$ is the measure of the interface between the cells linked to $x$ and $x_{i}$ in the Voronoi diagram (length in 2D, surface in 3D) and $d(x_{i})$, the distance between $x$ and $x_{i}$.

Discrete natural neighbor interpolation
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. A form of discrete natural neighbor interpolation has also been developed that gives a measure of interpolation uncertainty.

Properties
There are several useful properties of natural neighbor interpolation:


 * 1) The method is an exact interpolator, in that the original data values are retained at the reference data points.
 * 2) The method creates a smooth surface free from any discontinuities.
 * 3) The method is entirely local, as it is based on a minimal subset of data locations that excludes locations that, while close, are more distant than another location in a similar direction.
 * 4) The method is spatially adaptive, automatically adapting to local variation in data density or spatial arrangement.
 * 5) There is no requirement to make statistical assumptions.
 * 6) The method can be applied to very small datasets as it is not statistically based.
 * 7) The method is parameter free, so no input parameters that will affect the success of the interpolation need to be specified.