Polyrigid and polyaffine transformations: a novel geometrical tool to deal with non-rigid deformations - application to the registration of histological slices.

We describe in this paper a novel kind of geometrical transformations, named polyrigid and polyaffine. These transformations efficiently code for locally rigid or affine deformations with a small number of intuitive parameters. They can describe compactly large rigid or affine movements, unlike most free-form deformation classes. Very flexible, this tool can be readily adapted to a large variety of situations, simply by tuning the number of rigid or affine components and the number of parameters describing their regions of influence. The displacement of each spatial position is defined by a continuous trajectory that follows a differential equation which averages the influence of each rigid or affine component. We show that the resulting transformations are diffeomorphisms, smooth with respect to their parameters. We devise a new and flexible numerical scheme to allow a trade-off between computational efficiency and closeness to the ideal diffeomorphism. Our algorithms are implemented within the Insight Toolkit, whose generic programming style offers rich facilities for prototyping. In this context, we derive an effective optimization strategy of the transformations which demonstrates that this new tool is highly suitable for inference. The whole framework is exemplified successfully with the registration of histological slices. This choice is challenging, because these data often present locally rigid deformations added during their acquisition, and can also present a loss of matter, which makes their registration even more difficult. Powerful and flexible, this new tool opens up large perspectives, in non-rigid 3D rigid registration as well as in shape statistics.