Natural neighbor interpolation

Natural neighbor interpolation is a method of spatial interpolation, developed by Robin Sibson.[1] 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.

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.

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 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[2][3]

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.

See alsoEdit

ReferencesEdit

  1. ^ Sibson, R. (1981). "A brief description of natural neighbor interpolation (Chapter 2)". In V. Barnett (ed.). Interpreting Multivariate Data. Chichester: John Wiley. pp. 21–36.
  2. ^ N.H. Christ; R. Friedberg, R.; T.D. Lee (1982). "Weights of links and plaquettes in a random lattice". Nuclear Physics B. 210 (3): 337–346.
  3. ^ V.V. Belikov; V.D. Ivanov; V.K. Kontorovich; S.A. Korytnik; A.Y. Semenov (1997). "The non-Sibsonian interpolation: A new method of interpolation of the values of a function on an arbitrary set of points". Computational mathematics and mathematical physics. 37 (1): 9–15.

External linksEdit