Multivariate interpolation

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.

[1] A common special case is bivariate interpolation or two-dimensional interpolation, based on two variables or two dimensions.

The function to be interpolated is known at given points

and the interpolation problem consists of yielding values at arbitrary points

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).

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

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.

Catmull-Rom splines can be easily generalized to any number of dimensions.

The cubic Hermite spline article will remind you that

Rewrite this approximation as This formula can be directly generalized to N dimensions:[2] 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

-type operations for any type of tensor product splines, as explained in the tricubic interpolation article.

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.