View source for Multivariate interpolation

{{short description|Interpolation on functions of more than one variable}}
In [[Numerical analysis|numerical analysis]], '''multivariate interpolation''' is [[Interpolation|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 <math>(x_i, y_i, z_i, \dots)</math> and the interpolation problem consists of yielding values at arbitrary points <math>(x,y,z,\dots)</math>.

Multivariate interpolation is particularly important in [[Earth:Geostatistics|geostatistics]], where it is used to create a [[Earth:Digital elevation model|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 [[Earth:Hydrographic survey|hydrographic survey]]).

==Regular grid==
{{comparison_of_1D_and_2D_interpolation.svg|300px|}}
For function values known on a [[Regular grid|regular grid]] (having predetermined, not necessarily uniform, spacing), the following methods are available.

===Any dimension===
* [[Nearest-neighbor interpolation]]
* n-linear interpolation (see [[Bilinear interpolation|bi-]] and [[Trilinear interpolation|trilinear interpolation]] and [[Multilinear polynomial|multilinear polynomial]])
* n-cubic interpolation (see [[Bicubic interpolation|bi-]] and [[Tricubic interpolation|tricubic interpolation]])
* [[Kriging]]
* [[Inverse distance weighting]]
* [[Natural neighbor interpolation]]
* [[Spline interpolation]]
* [[Radial basis function interpolation]]

===2 dimensions===
* [[Barnes interpolation]]
* [[Bilinear interpolation]]
* [[Bicubic interpolation]] 
* [[Bézier surface]]
* [[Lanczos resampling]] 
* [[Delaunay triangulation]]

Bitmap resampling is the application of 2D multivariate interpolation in [[Image processing|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.
{{Gallery
|title=
|width=300px
  |File:Interpolation-nearest.svg|Nearest neighbor
  |File:Interpolation-bilinear.svg|Bilinear
  |File:Interpolation-bicubic.svg|Bicubic
}}

See also [[Padua points]], for [[Polynomial interpolation|polynomial interpolation]] in two variables.

===3 dimensions===
* [[Trilinear interpolation]]
* [[Tricubic interpolation]]

See also bitmap resampling.

===Tensor product splines for ''N'' dimensions===

Catmull-Rom splines can be easily generalized to any number of dimensions.
The [[Cubic Hermite spline|cubic Hermite spline]] article will remind you that <math>\mathrm{CINT}_x(f_{-1}, f_0, f_1, f_2) = \mathbf{b}(x) \cdot \left( f_{-1} f_0 f_1 f_2 \right)</math> for some 4-vector <math>\mathbf{b}(x)</math> which is a function of ''x'' alone, where <math>f_j</math> is the value at <math>j</math> of the function to be interpolated.
Rewrite this approximation as
:<math>
\mathrm{CR}(x) = \sum_{i=-1}^2 f_i b_i(x)
</math>
This formula can be directly generalized to N dimensions:<ref>[https://arxiv.org/abs/0905.3564 Two hierarchies of spline interpolations. Practical algorithms for multivariate higher order splines]</ref>
:<math>
\mathrm{CR}(x_1,\dots,x_N) = \sum_{i_1,\dots,i_N=-1}^2 f_{i_1\dots i_N} \prod_{j=1}^N b_{i_j}(x_j)
</math>
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 <math>\mathrm{CINT}</math>-type operations for any type of tensor product splines, as explained in the [[Tricubic interpolation|tricubic interpolation]] article.
However, the fact remains that if there are <math>n</math> terms in the 1-dimensional <math>\mathrm{CR}</math>-like summation, then there will be <math>n^N</math> terms in the <math>N</math>-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|linear interpolation]] (a type of [[Piecewise linear function|piecewise linear function]])
** n-[[Simplex|simplex]] (e.g. tetrahedron) interpolation (see [[Barycentric coordinate system|barycentric coordinate system]])
* [[Inverse distance weighting]]
* [[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 spline]]s are a special case of radial basis functions with low degree polynomial terms)
* Least-squares [[Spline (mathematics)|spline]]
* Natural neighbour interpolation
{{anchor|Gridding}}''Gridding'' is the process of converting irregularly spaced data to a regular grid (gridded data).

==See also==
*[[Smoothing]]
*Surface fitting

==Notes==
<references />

==External links==
* [http://chichi.lalescu.ro/splines.html Example C++ code for several 1D, 2D and 3D spline interpolations (including Catmull-Rom splines).]
* [https://web.archive.org/web/20060915111500/http://www.ices.utexas.edu/CVC/papers/multidim.pdf Multi-dimensional Hermite Interpolation and Approximation], Prof. Chandrajit Bajaja, [[Organization:Purdue University|Purdue University]]
* [https://github.com/DurhamDecLab/ARBInterp Python library containing 3D and 4D spline interpolation methods.]

[[Category:Multivariate interpolation| ]]
[[Category:Interpolation]]

{{Sourceattribution|Multivariate interpolation}}