Trilinear interpolation


Trilinear interpolation is a method of multivariate interpolation on a 3-dimensional regular grid. It approximates the value of a function at an intermediate point within the local axial rectangular prism linearly, using function data on the lattice points. For an arbitrary, unstructured mesh, other methods of interpolation must be used; if all the mesh elements are tetrahedra, then barycentric coordinates provide a straightforward procedure.
Trilinear interpolation is frequently used in numerical analysis, data analysis, and computer graphics.

Compared to linear and bilinear interpolation

Trilinear interpolation is the extension of linear interpolation, which operates in spaces with dimension, and bilinear interpolation, which operates with dimension, to dimension. These interpolation schemes all use polynomials of order 1, giving an accuracy of order 2, and it requires adjacent pre-defined values surrounding the interpolation point. There are several ways to arrive at trilinear interpolation, which is equivalent to 3-dimensional tensor B-spline interpolation of order 1, and the trilinear interpolation operator is also a tensor product of 3 linear interpolation operators.

Method

On a periodic and cubic lattice, let,, and
be the differences between each of,, and the smaller coordinate related, that is:
where indicates the lattice point below, and indicates the lattice point above and similarly for
and.
First we interpolate along , giving:
Where means the function value of Then we interpolate these values, giving:
Finally we interpolate these values along :
This gives us a predicted value for the point.
The result of trilinear interpolation is independent of the order of the interpolation steps along the three axes: any other order, for instance along, then along, and finally along, produces the same value.
The above operations can be visualized as follows: First we find the eight corners of a cube that surround our point of interest. These corners have the values,,,,,,,.
Next, we perform linear interpolation between and to find, and to find, and to find, and to find.
Now we do interpolation between and to find, and to find. Finally, we calculate the value via linear interpolation of and
In practice, a trilinear interpolation is identical to two bilinear interpolation combined with a linear interpolation:

Alternative algorithm

An alternative way to write the solution to the interpolation problem is
where the coefficients are found by solving the linear system
yielding the result