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 .
First we interpolate along (imagine we are pushing the front face of the cube to the back), giving:
Where means the function value of Then we interpolate these values (along, as we were pushing the top edge to the bottom), giving:
Finally we interpolate these values along (walking through a line):
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 C000, C100, C010, C110, C001, C101, C011, C111.
Next, we perform linear interpolation between C000 and C100 to find C00, C001 and C101 to find C01, C011 and C111 to find C11, C010 and C110 to find C10.
Now we do interpolation between C00 and C10 to find C0, C01 and C11 to find C1. Finally, we calculate the value C via linear interpolation of C0 and C1
In practice, a trilinear interpolation is identical to three successive linear interpolations, or two bilinear interpolations combined with a linear interpolation:
Read more about this topic: Trilinear Interpolation
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