The concepts and arguments introduced in this section show why there
is an ever-increasing interest in non-rigid registration, based on
non-rigid transformations^{26}. The mathematics behind the required transformations and the theory
that needs to be established in order to make them practical is constantly
being explored and papers on the subject receive attention and recognition.
Diffeomorphic functions are *invertible, continuous* and *one-to-one*
mappings for a given image^{27}. These mappings are usually described by some local geometrical transformations
that have an effect on pixels or the plane that pixels are embedded
in.

Current transformations that are used in Manchester University by Twining and Marsland [28] also benefit from having continuous derivatives at the boundaries, unlike for example, Lötjönen and Mäkelä [29] who suggested a similar transformation. This, however, is a convenient property that is not a necessity. It is just a strategically good attribute to have in real-world applications.

What invertibility, continuity and one-to-one mappings mean in simpler terms is that for each transformation:

- That transformation has an inverse so that any transformation (or
*warp*as it will be later referred to as) can be reversed. - That transformation affects
*all*data (pixels) within its boundaries so it has a centralised yet somewhat global effect^{28}. This means that every point must move as would be expected to give a continuous flow of intensities. The transformation should also be cautious not to corrupt structures in the image in any way. - No two points should be mapped onto the same point as this would ``strip off'' areas in the image.