In numerical analysis, multivariate interpolation or multidimensional interpolation is interpolation on multivariate functions, having more than one variable or defined over a multi-dimensional domain. A common special case is bivariate interpolation or two-dimensional interpolation, based on two variables or two dimensions. 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 , … ) {\displaystyle (x_{i},y_{i},z_{i},\dots )} and the interpolation problem consists of yielding values at arbitrary points ( x , y , z , … ) {\displaystyle (x,y,z,\dots )} . 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 Barnes interpolation Bilinear interpolation Bicubic interpolation Bézier surface Lanczos resampling Delaunay triangulation 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 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 article will remind you that C I N T x ( f − 1 , f 0 , f 1 , f 2 ) = b ( x ) ⋅ ( f − 1 f 0 f 1 f 2 ) {\displaystyle \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)} for some 4-vector b ( x ) {\displaystyle \mathbf {b} (x)} which is a function of x alone, where f j {\displaystyle f_{j}} is the value at j {\displaystyle j} of the function to be interpolated. Rewrite this approximation as
C R ( x ) = ∑ i = − 1 2 f i b i ( x ) {\displaystyle \mathrm {CR} (x)=\sum _{i=-1}^{2}f_{i}b_{i}(x)}
This formula can be directly generalized to N dimensions:
C R ( x 1 , … , x N ) = ∑ i 1 , … , i N = − 1 2 f i 1 … i N ∏ j = 1 N b i j ( x j ) {\displaystyle \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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