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.^[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.

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^[2]^[3]: $w_{i}(\mathbf {x} )={\frac {\frac {l(\mathbf {x} _{i})}{d(\mathbf {x} _{i})}}{\sum _{k=1}^{n}{\frac {l(\mathbf {x} _{k})}{d(\mathbf {x} _{k})}}}}$

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 $x i$ .

References

↑ 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.
↑ 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.
↑ 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 links

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[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.

[1]

[2]

[3]

See also

References

External links