The concepts and arguments introduced so far in this chapter show why there is an ever-increasing interest in non-rigid registration, based on non-rigid transformations3.9. 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 image3.10. 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 diffeomorphic transformations that are used in Manchester University by Twining and Marsland [] also benefit from having continuous derivatives at the boundaries, unlike for example, these of Lötjönen and Mäkelä [] who suggested a similar transformation type. 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: