OverviewMultivariate interpolation is an area of data fitting which, as opposed to univariate interpolation which fitted two-dimensional data points, finds the surface that provides an exact fit to a series of multidimensional data points. It is called multivariate since the data points are supposed to be sampled from a function of several variables. Formally speaking, consider a series of distinct - dimensional data points , , where is a vector, for each . By interpolating these data points we mean finding a function such that:
Nearest-neighbor InterpolationThis type of interpolation basically assigns to any point in the plane, the value of the closest data point to . Formally, given a series of data points , for , the corresponding nearest-neighbor interpolation function is given by
In general d-dimensional space, nearest-neighbor interpolation assigns to some point the value of the closest data point to , i.e. the one which minimizes the objective function
Bilinear InterpolationThis is a generalization of linear interpolation, from 2D to 3D data points. It is assumed that the given data points are distributed along an uniform grid, as are the points , , and in the image below. The aim is to estimate the value of the function (from which the data points have been sampled) at point in the above graph. In order to do this, let us define two auxiliary functions through
This solves the problem of doing bilinear interpolation for a set of 4 three-dimensional points. If there are more than 4 points (they should however be a multiple of 2), then we repeat the above algorithm for each cell. The interpolation function over the entire domain is then defined in a piecewise manner on each cell, through the corresponding bilinear interpolation function for that cell.The image below shows the values obtained by applying linear interpolation on the same series of data points as in the previous graph.
Using the same reasoning as above, we are able to generalize linear interpolation from some -dimensional space to -dimensional space, in a recursive manner, giving birth to multilinear interpolation. The concept of uniform grid also generalizes to multidimensional space, as seen in the article http://en.wikipedia.org/wiki/Honeycomb_(geometry) .