Trilineare Interpolation

Voxels | | DICOM

For medical CT and MR volume data the function values (that is scalar density) $\mu_{ijk}$ are given at the corner points $P_{ijk}$ of a voxel (rot).

Tri-linear interpolation creates a single interpolated value (blue) from 8 neighbouring scalar density values (red) by three subsequent linear interpolations along the three axis with respective normalized interpolation weights $u,v$ and $w$.

Linear interpolation weights $u,v,w\in[0,1]$:

$\displaystyle{ \mu(u,v,w) = }$
$\displaystyle{ (1-w)((1-v)((1-u)\mu_{000}+u\mu_{100}) + }$
$\displaystyle{ v((1-u)\mu_{010}+u\mu_{110})) + }$
$\displaystyle{ w((1-v)((1-u)\mu_{001}+u\mu_{101}) + }$
$\displaystyle{ v((1-u)\mu_{011}+u\mu_{111})) }$

Voxels | | DICOM