image:Natural-neighbors-coefficients-example.png, 200px, 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 diagram, Voronoi tessellation of a discrete set of spatial points. This has advantages over simpler methods of interpolation, such as
nearest-neighbor interpolation
Nearest-neighbor interpolation (also known as proximal interpolation or, in some contexts, point sampling) is a simple method of multivariate interpolation in one or more dimensions.
Interpolation is the problem of approximating the value of ...
, in that it provides a smoother approximation to the underlying "true" function.
The basic equation is:
:
where
is the estimate at
,
are the weights and
are the known data at
. The weights,
, are calculated by finding how much of each of the surrounding areas is "stolen" when inserting
into the tessellation.
;Sibson weights
:
where is the volume of the new cell centered in , and is the volume of the intersection between the new cell centered in and the old cell centered in .
200px, Natural neighbor interpolation with Laplace weights. The interface between the cells linked to and is in blue, while the distance between and is in red.
;Laplace weights
:
where is the
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
of the interface between the cells linked to and in the
Voronoi diagram
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed th ...
(length in 2D, surface in 3D) and , the distance between and .
See also
*
Inverse distance weighting
Inverse distance weighting (IDW) is a type of deterministic method for multivariate interpolation with a known scattered set of points. The assigned values to unknown points are calculated with a weighted average of the values available at the kno ...
*
Multivariate interpolation
In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable; when the variates are spatial coordinates, it is also known as spatial interpolation.
The function to be interpolated is known at given poin ...
References
External links
Natural Neighbor InterpolationImplementation notes for natural neighbor, and comparison to other interpolation methods*
ttps://github.com/innolitics/natural-neighbor-interpolation Fast, discrete natural neighbor interpolation in 3D on the CPU
Multivariate interpolation
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