Multivariate interpolation

• 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

• 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.

• Trilinear interpolation
• Tricubic interpolation

See also bitmap resampling.

Catmull-Rom splines can be easily generalized to any number of dimensions. The cubic Hermite spline article will remind you that ${\displaystyle \mathrm {CINT} _{x}(f_{-1},f_{0},f_{1},f_{2})=\mathbf {b} (x)\cdot \left(f_{-1}f_{0}f_{1}f_{2}\right)}$ for some 4-vector ${\displaystyle \mathbf {b} (x)}$ which is a function of x alone, where ${\displaystyle f_{j}}$ is the value at ${\displaystyle j}$ of the function to be interpolated. Rewrite this approximation as

${\displaystyle \mathrm {CR} (x)=\sum _{i=-1}^{2}f_{i}b_{i}(x)}$

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

${\displaystyle \mathrm {CR} (x_{1},\dots ,x_{N})=\sum _{i_{1},\dots ,i_{N}=-1}^{2}f_{i_{1}\dots i_{N}}\prod _{j=1}^{N}b_{i_{j}}(x_{j})}$

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 ${\displaystyle \mathrm {CINT} }$-type operations for any type of tensor product splines, as explained in the tricubic interpolation article. However, the fact remains that if there are ${\displaystyle n}$ terms in the 1-dimensional ${\displaystyle \mathrm {CR} }$-like summation, then there will be ${\displaystyle n^{N}}$ terms in the ${\displaystyle N}$-dimensional summation.