# Natural neighbor interpolation Natural neighbor interpolation with Sibson weights. The area of the green circles are the interpolating weights, wi. The purple-shaded region is the new Voronoi cell, after inserting the point to be interpolated (black dot). The weights represent the intersection areas of the purple-cell with each of the seven surrounding cells.

Natural neighbor 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:

$G(x)=\sum _{i=1}^{n}{w_{i}(x)f(x_{i})}$ 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(xi) is the volume of the intersection between the new cell centered in x and the old cell centered in xi. Natural neighbor interpolation with Laplace weights. The interface l(xi) between the cells linked to x and xi is in blue, while the distance d(xi) between x and xi is in red.
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(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.