Natural neighbor interpolation
# Natural neighbor interpolation

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 $A(x_i)$ 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 $x_i.$

