The alternate method that is usually preferable
to looping over input space
is that
our program have a loop over the space of the outputs,
and that each output find its input.
The matrix multiply of (4.2)
can be interpreted this way.
Where the transformation shrinks is a small problem.
In that area
many points in the input space are ignored,
where perhaps they should somehow be averaged with their neighbors.
This is not a serious problem unless we are contemplating
iterative transformations back and forth between the spaces.
We will now address interesting questions
about
the reversibility of these deformation transforms.