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| In [[numerical analysis]], '''multivariate interpolation''' or '''spatial interpolation''' is [[interpolation]] on functions of more than one variable.
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| The function to be interpolated is known at given points <math>(x_i, y_i, z_i, \dots)</math> and the interpolation problem consist of yielding values at arbitrary points <math>(x,y,z,\dots)</math>.
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| ==Regular grid==
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| For function values known on a [[regular grid]] (having predetermined, not necessarily uniform, spacing), the following methods are available. | |
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| ===Any dimension===
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| * [[Nearest-neighbor interpolation]]
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| ===2 dimensions===
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| * [[Barnes interpolation]]
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| * [[Bilinear interpolation]]
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| * [[Bicubic interpolation]]
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| * [[Bézier surface]]
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| * [[Lanczos resampling]]
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| * [[Delaunay triangulation]]
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| * [[Inverse distance weighting]]
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| * [[Kriging]]
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| * [[Natural neighbor]]
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| * [[Spline interpolation]]
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| [[Resampling (bitmap)|Bitmap resampling]] is the application of 2D multivariate interpolation in [[image processing]].
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| Three of the methods applied on the same dataset, from 16 values located at the black dots. The colours represent the interpolated values.
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| <gallery>
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| Image:Nearest2DInterpolExample.png|Nearest neighbor
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| Image:BilinearInterpolExample.png|Bilinear
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| Image:BicubicInterpolationExample.png|Bicubic
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| </gallery>
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| See also [[Padua points]], for [[polynomial interpolation]] in two variables.
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| ===3 dimensions===
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| * [[Trilinear interpolation]]
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| * [[Tricubic interpolation]]
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| See also [[Resampling (bitmap)|bitmap resampling]].
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| ===Tensor product splines for ''N'' dimensions===
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| Catmull-Rom splines can be easily generalized to any number of dimensions.
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| The [[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.
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| Rewrite this approximation as
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| :<math> | |
| \mathrm{CR}(x) = \sum_{i=-1}^2 f_i b_i(x)
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| </math>
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| This formula can be directly generalized to N dimensions:<ref>[http://arxiv.org/abs/0905.3564 Two hierarchies of spline interpolations. Practical algorithms for multivariate higher order splines]</ref>
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| :<math>
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| \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)
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| </math>
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| Note that similar generalizations can be made for other types of spline interpolations, including Hermite splines.
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| 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]] article.
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| 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.
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| == Irregular grid (scattered data) ==
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| Schemes defined for scattered data on an [[irregular grid]] should all work on a regular grid, typically reducing to another known method.
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| * [[Nearest-neighbor interpolation]]
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| * [[Triangulated irregular network]]-based [[natural neighbor]]
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| * [[Triangulated irregular network]]-based [[linear interpolation]] (a type of [[piecewise linear function]])
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| * [[Inverse distance weighting]]
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| * [[Kriging]]
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| * [[Radial basis function]]
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| * [[Thin plate spline]]
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| * [[Polyharmonic spline]] (the thin-plate-spline is a special case of a polyharmonic spline)
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| * Least-squares [[spline (mathematics)|spline]]
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| ==Notes==
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| <references />
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| ==External links==
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| * [http://chichi.lalescu.ro/splines.html Example C++ code for several 1D, 2D and 3D spline interpolations (including Catmull-Rom splines).]
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| * [http://web.archive.org/web/20060915111500/http://www.ices.utexas.edu/CVC/papers/multidim.pdf Multi-dimensional Hermite Interpolation and Approximation], Prof. Chandrajit Bajaja, [[Purdue University]]
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| [[Category:Interpolation]]
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| [[Category:Multivariate interpolation| ]]
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