Natural-neighbor interpolation

Natural-neighbor interpolation or Sibson 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.

Formulation

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 $x i$ .

Properties

There are several useful properties of natural neighbor interpolation:^[4]

The method is an exact interpolator, in that the original data values are retained at the reference data points.
The method creates a smooth surface free from any discontinuities.
The method is entirely local, as it is based on a minimal subset of data locations that excludes locations that, while close, are more distant than another location in a similar direction.
The method is spatially adaptive, automatically adapting to local variation in data density or spatial arrangement.
There is no requirement to make statistical assumptions.
The method can be applied to very small datasets as it is not statistically based.
The method is parameter free, so no input parameters that will affect the success of the interpolation need to be specified.

Extensions

Natural neighbor interpolation has also been implemented in a discrete form, which has been demonstrated to be computationally more efficient in at least some circumstances.^[5] A form of discrete natural neighbor interpolation has also been developed that gives a measure of interpolation uncertainty.^[4]

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. Bibcode:1982NuPhB.210..337C. doi:10.1016/0550-3213(82)90124-9.
^ 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.
^ ^a ^b Etherington, Thomas R. (2020-07-13). "Discrete natural neighbour interpolation with uncertainty using cross-validation error-distance fields". PeerJ Computer Science. 6: e282. doi:10.7717/peerj-cs.282. ISSN 2376-5992. PMC 7924714. PMID 33816933. This article incorporates text available under the CC BY 4.0 license.
^ Park, S.W.; Linsen, L.; Kreylos, O.; Owens, J.D.; Hamann, B. (2006). "Discrete Sibson interpolation". IEEE Transactions on Visualization and Computer Graphics. 12 (2): 243–253. doi:10.1109/TVCG.2006.27. PMID 16509383.

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. Bibcode:1982NuPhB.210..337C. doi:10.1016/0550-3213(82)90124-9.

[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.

[:0-4] Etherington, Thomas R. (2020-07-13). "Discrete natural neighbour interpolation with uncertainty using cross-validation error-distance fields". PeerJ Computer Science. 6: e282. doi:10.7717/peerj-cs.282. ISSN 2376-5992. PMC 7924714. PMID 33816933. This article incorporates text available under the CC BY 4.0 license.

[5] Park, S.W.; Linsen, L.; Kreylos, O.; Owens, J.D.; Hamann, B. (2006). "Discrete Sibson interpolation". IEEE Transactions on Visualization and Computer Graphics. 12 (2): 243–253. doi:10.1109/TVCG.2006.27. PMID 16509383.

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