Transformations

A transformation from one volume to another can act either in voxel coordinates, world coordinates or a combination of both. In general, voxel and world coordinates are related by a simple affine transformation $T_W : V \rightarrow W$, where $V$ is the voxel coordinate space, and $W$ the world coordinate space. By using this transformation, all transformations can be expressed as transformations between world coordinates, which is more convenient and will be assumed from here on.

For example, if $T : V_2 \rightarrow V_1$ is a transformation between two voxel coordinate spaces, then $T_{W1}^{-1} \, T \, T_{W2}$ is the corresponding transformation between the world coordinate spaces.