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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. A form of discrete natural neighbor interpolation has also been developed that gives a measure of interpolation uncertainty.
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43:. 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.
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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".
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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.
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740:{\displaystyle 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})}}}}}
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The method is parameter free, so no input parameters that will affect the success of the interpolation need to be specified.
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The method is spatially adaptive, automatically adapting to local variation in data density or spatial arrangement.
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Natural neighbor interpolation with Sibson weights. The area of the green circles are the interpolating weights,
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The method is an exact interpolator, in that the original data values are retained at the reference data points.
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Sibson, R. (1981). "A brief description of natural neighbor interpolation (Chapter 2)". In V. Barnett (ed.).
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969:"Discrete natural neighbour interpolation with uncertainty using cross-validation error-distance fields"
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of a discrete set of spatial points. This has advantages over simpler methods of interpolation, such as
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N.H. Christ; R. Friedberg, R.; T.D. Lee (1982). "Weights of links and plaquettes in a random lattice".
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Implementation notes for natural neighbor, and comparison to other interpolation methods
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The method can be applied to very small datasets as it is not statistically based.
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Interactive
Voronoi diagram and natural neighbor interpolation visualization
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There are several useful properties of natural neighbor interpolation:
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Park, S.W.; Linsen, L.; Kreylos, O.; Owens, J.D.; Hamann, B. (2006).
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Natural neighbor interpolation with
Laplace weights. The interface
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is the volume of the intersection between the new cell centered in
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The method creates a smooth surface free from any discontinuities.
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Fast, discrete natural neighbor interpolation in 3D on the CPU
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There is no requirement to make statistical assumptions.
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167:{\displaystyle G(x)=\sum _{i=1}^{n}{w_{i}(x)f(x_{i})}}
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