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Diffeomorphism

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 [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:

1. That transformation has an inverse so that any transformation (or warp as it will be later referred to as) can be reversed.
2. That transformation affects all data (pixels) within its boundaries so it has a centralised yet somewhat global effect²⁸. 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.
3. No two points should be mapped onto the same point as this would ``strip off'' areas in the image.