Concept

# Multivariate interpolation

Summary
In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable (multivariate functions); when the variates are spatial coordinates, it is also known as spatial interpolation. 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. 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 . 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 for some 4-vector which is a function of x alone, where is the value at of the function to be interpolated. Rewrite this approximation as This formula can be directly generalized to N dimensions: Note that similar generalizations can be made for other types of spline interpolations, including Hermite splines.