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Warps Representation

The potential of non-rigid registration of images has been a subject of research due to its ability to simplify the correspondence problem, amongst some other advantages it offers. As opposed to translation, scaling and rotation, all of which are rigid transformations, affine and non-rigid transformations [12] cater for flexible manipulation of points of interest. Folding and tearing has been the main drawback algebraic implementations of these transformation, but recently an interesting group of warps has been investigated. So-called diffeomorphic warps offer a solution to this drawback and they can easily be extended to 3-D in their regular form as shown in the figure below. Nevertheless, for most practical uses, a large number of such warps is needed, resulting in high computational demand. For further discussion of the application of non-rigid registration to landmark selection, see the work described in Hill et al.[13] and Rueckert et al. [14].

Figure 3 illustrates the effects current type of warps have on the space used to embed images. These warps are reminiscent of the ones described in Lötjönen and Mäkelä [15], but unlike many others, they have continuous derivates at the borders, which is a crucial condition for diffeomorphism.

Figure 3: Warp example

When dealing with the aforementioned appearance models, an alternative emerges which chooses to deal with similarity measures using warps that minimise the difference between two images. Current research work attempts to apply the same principles to a large group of images and the result is a parameterisation that is compact in a global context. It relies on the many warps applied to the input data which bring their collective descriptive parameters closer together. As the different images are embedded in the heavily warped space, the spatial differences amongst the images are essentially being minimised.