Multivariate interpolation explained

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

(x_i,y_i,z_i,...)

and the interpolation problem consists of yielding values at arbitrary points

(x,y,z,...)

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).

Regular grid

For function values known on a regular grid (having predetermined, not necessarily uniform, spacing), the following methods are available.

Any dimension

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

2 dimensions

Bitmap resampling is the application of 2D multivariate interpolation in image processing.

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.

3 dimensions

Tensor product splines for N dimensions

Catmull-Rom splines can be easily generalized to any number of dimensions.The cubic Hermite spline article will remind you that

CINT_x(f_-1,f_0,f_1,f₂₎=b(x) ⋅ \left(f_-1f₀f₁f₂\right)

for some 4-vector

b(x)

which is a function of x alone, where

f_j

is the value at

of the function to be interpolated.Rewrite this approximation as

CR(x)=

	2
\sum
	i=-1

f_ib_i(x)

This formula can be directly generalized to N dimensions:^[1]

CR(x_1,...,x_N)=

	2
\sum
	i_1,...,i_N=-1

f
	i_1...i_N

	N
\prod
	j=1

b
	i_j

(x_j)

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

CINT

-type operations for any type of tensor product splines, as explained in the tricubic interpolation article.However, the fact remains that if there are

terms in the 1-dimensional

-like summation, then there will be

n^N

terms in the

-dimensional summation.

Irregular grid (scattered data)

Schemes defined for scattered data on an irregular grid are more general.They should all work on a regular grid, typically reducing to another known method.

Nearest-neighbor interpolation
Triangulated irregular network-based natural neighbor
Triangulated irregular network-based linear interpolation (a type of piecewise linear function)
- n-simplex (e.g. tetrahedron) interpolation (see barycentric coordinate system)
Inverse distance weighting
ABOS - approximation based on smoothing
Kriging
Gradient-enhanced kriging (GEK)
Thin plate spline
Polyharmonic spline (the thin-plate-spline is a special case of a polyharmonic spline)
Radial basis function (Polyharmonic splines are a special case of radial basis functions with low degree polynomial terms)
Least-squares spline
Natural neighbour interpolation

Gridding is the process of converting irregularly spaced data to a regular grid (gridded data).

Notes

https://arxiv.org/abs/0905.3564 Two hierarchies of spline interpolations. Practical algorithms for multivariate higher order splines