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Concept# Multivariate interpolation

Summary

In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable (multivariate functions); when the variates are spatial coordinates, it is also known as spatial interpolation.
The function to be interpolated is known at given points and the interpolation problem consists of yielding values at arbitrary points .
Multivariate interpolation is particularly important in geostatistics, where it is used to create a digital elevation model from a set of points on the Earth's surface (for example, spot heights in a topographic survey or depths in a hydrographic survey).
For function values known on a regular grid (having predetermined, not necessarily uniform, spacing), the following methods are available.
Nearest-neighbor interpolation
n-linear interpolation (see bi- and trilinear interpolation and multilinear polynomial)
n-cubic interpolation (see bi- and tricubic interpolation)
Kriging
Inverse distance weighting
Natural neighbor interpolation
Spline interpolation
Radial basis function interpolation
Barnes interpolation
Bilinear interpolation
Bicubic interpolation
Bézier surface
Lanczos resampling
Delaunay triangulation
Bitmap resampling is the application of 2D multivariate interpolation in .
Three of the methods applied on the same dataset, from 25 values located at the black dots. The colours represent the interpolated values.
See also Padua points, for polynomial interpolation in two variables.
Trilinear interpolation
Tricubic interpolation
See also bitmap resampling.
Catmull-Rom splines can be easily generalized to any number of dimensions.
The cubic Hermite spline article will remind you that for some 4-vector which is a function of x alone, where is the value at of the function to be interpolated.
Rewrite this approximation as
This formula can be directly generalized to N dimensions:
Note that similar generalizations can be made for other types of spline interpolations, including Hermite splines.

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Bicubic interpolation

In mathematics, bicubic interpolation is an extension of cubic spline interpolation (a method of applying cubic interpolation to a data set) for interpolating data points on a two-dimensional regular grid. The interpolated surface (meaning the kernel shape, not the image) is smoother than corresponding surfaces obtained by bilinear interpolation or nearest-neighbor interpolation. Bicubic interpolation can be accomplished using either Lagrange polynomials, cubic splines, or cubic convolution algorithm.

Multivariate interpolation

In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable (multivariate functions); when the variates are spatial coordinates, it is also known as spatial interpolation. The function to be interpolated is known at given points and the interpolation problem consists of yielding values at arbitrary points . Multivariate interpolation is particularly important in geostatistics, where it is used to create a digital elevation model from a set of points on the Earth's surface (for example, spot heights in a topographic survey or depths in a hydrographic survey).

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