Looping over output space

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 (11) 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.