Multivariate interpolation

In numerical analysis, multivariate interpolation or spatial interpolation is interpolation on functions of more than one variable.
The function to be interpolated is known at given points and the interpolation problem consist of yielding values at arbitrary points .
Contents
Regular grid
For function values known on a regular grid (having predetermined, not necessarily uniform, spacing), the following methods are available.
Any dimension
2 dimensions
 Bilinear interpolation
 Bicubic interpolation
 Bézier surface
 Lanczos resampling
 Delaunay triangulation
 Spline interpolation
 Natural neighbor
 Kriging
 Inverse distance weighting
Bitmap resampling is the application of 2D multivariate interpolation in image processing.
Three of the methods applied on the same dataset, from 16 values located at the black dots. The colours represent the interpolated values.
See also Padua points, for polynomial interpolation in two variables.
3 dimensions
See also bitmap resampling.
Tensor product splines for N dimensions
CatmullRom splines can be easily generalized to any number of dimensions. The cubic Hermite spline article will remind you that for some 4vector which is a function of x alone, where f_{j} is the value at j of the function to be interpolated. Rewrite this approximation as
This formula can be directly generalized to N dimensions ^{[1]}:
Note that similar generalizations can be made for other types of spline interpolations, including Hermite splines. In regards to efficiency, the general formula can in fact be computed as a composition of successive CINTtype operations for any type of tensor product splines, as explained in the tricubic interpolation article. However, the fact remains that if there are n terms in the 1dimensional CRlike summation, then there will be n^{N} terms in the Ndimensional summation.
Irregular grid (scattered data)
Schemes defined for scattered data on a irregular grid should all work on a regular grid, typically reducing to another known method.
 Nearestneighbor interpolation
 Triangulated irregular networkbased natural neighbor
 Triangulated irregular networkbased linear interpolation (a type of piecewise linear function)
 Inverse distance weighting
 Kriging
 Radial basis function
 Thin plate spline
 Polyharmonic spline (the thinplatespline is a special case of a polyharmonic spline)
 Leastsquares spline
Notes
 ^ Two hierarchies of spline interpolations. Practical algorithms for multivariate higher order splines
External links
 Example C++ code for several 1D, 2D and 3D spline interpolations (including CatmullRom splines).
 Multidimensional Hermite Interpolation and Approximation, Prof. Chandrajit Bajaja, Purdue University
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