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There are important factors to consider when selecting a transformation method. One such factor is the property called diffeomorphism. Diffeomorphic [] functions are invertible, continuous and one-to-one mappings, which can be applied to a given image. Diffeomorphic transformations that are used in this work were initially devised by Twining and Marsland [] (see the example in Figure ). These benefit from having continuous derivatives at the boundaries unlike, for example, those proposed by Lötjönen and Mäkelä [].

Figure: An example of image warping in medical contexts (the human brain). Points that are overlaid on the skull depict knot-points for the splines that render a transformation, which is based on clamped-plate splines. The image on the right is a warped and tilted version of the original on the left.

Figure: A pseudo-non-rigid warp example. The effect of the warp on a normal grid (left) is shown on the right hand side. Because lines in the grid do not intersect by collapsing onto one another, the warp is considered to be diffeomorphic.

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

1. The transformation has a calculable inverse transformation. This way, any transformation can in principle be reversed, i.e. its effect retracted.
2. The transformation affects all data (e.g. image pixels) within its boundaries so it has a spatially-contained effect^2.3. This means that every point must move as would be expected to give a continuous flow of intensities.
3. No two points should be mapped onto the same point as this would `strip off' areas of the image, depleting them from data.