Natural-neighbor interpolation

Natural neighbor interpolation with Sibson weights. The area of the green circles are the interpolating weights, w_i. 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(x_i) is the volume of the intersection between the new cell centered in x and the old cell centered in x_i.

Natural neighbor interpolation with Laplace weights. The interface l(x_i) between the cells linked to x and x_i is in blue, while the distance d(x_i) between x and x_i 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(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.

See also

• Inverse distance weighting
• Multivariate interpolation

External links

• Natural Neighbor Interpolation
• Implementation notes for natural neighbor, and comparison to other interpolation methods
• Interactive Voronoi diagram and natural neighbor interpolation visualization
• Fast, discrete natural neighbor interpolation in 3D on the CPU