Multivariate interpolation#Gridding

{{comparison_of_1D_and_2D_interpolation.svg|300px|}}

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

• 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.

{{Gallery

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|File:Interpolation-nearest.svg|Nearest neighbor

|File:Interpolation-bilinear.svg|Bilinear

|File:Interpolation-bicubic.svg|Bicubic

}}

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 $\backslash mathrm\{CINT\}\_x(f\_\{-1\},\; f\_0,\; f\_1,\; f\_2)\; =\; \backslash mathbf\{b\}(x)\; \backslash cdot\; \backslash left(\; f\_\{-1\}\; f\_0\; f\_1\; f\_2\; \backslash right)$ for some 4-vector $\backslash mathbf\{b\}(x)$ which is a function of x alone, where $f\_j$ is the value at $j$ of the function to be interpolated.

Rewrite this approximation as

:$$

\mathrm{CR}(x) = \sum_{i=-1}^2 f_i b_i(x)

This formula can be directly generalized to N dimensions:[https://arxiv.org/abs/0905.3564 Two hierarchies of spline interpolations. Practical algorithms for multivariate higher order splines]

:$$

\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 $\backslash mathrm\{CINT\}$-type operations for any type of tensor product splines, as explained in the tricubic interpolation article.

However, the fact remains that if there are $n$ terms in the 1-dimensional $\backslash mathrm\{CR\}$-like summation, then there will be $n^N$ terms in the $N$-dimensional summation.

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
• [https://surgeweb.mzf.cz/AOSIM/ABOS.htm 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

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

• Smoothing
• Surface fitting

• [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, Purdue University
• [https://github.com/DurhamDecLab/ARBInterp Python library containing 3D and 4D spline interpolation methods.]

Category:Interpolation