The concepts and arguments introduced in this section show why there is an ever-increasing interest in non-rigid registration, based on non-rigid transformations26. 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 image27. 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  also benefit from having continuous derivatives at the boundaries, unlike for example, Lötjönen and Mäkelä  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: