Laurent Younes:


Riemannian geometry of deformable templates

A collection of deformable objets, like shapes (represented as curves, or as configurations of points), images, 3D volumes... can be, from an abstract point of view, represented as a Riemannian manifold (the objects) on which a (preferably Lie) group is acting (the deformations). Even if the action is smooth (which is not even obvious when the number of dimensions is infinite), the prior metric on the manifold may fail to correctly comply with the action, in the sense that small deformations may result in too large variations for this metric. We here provide a simple device in which the initial metric is modified in a (so-called) procrustean manner, to obtain a new Riemannian structure which now behaves correctly with respect to the group action.

This approach will be applied to landmark matching interpolation, in which the problem is to estimate a dense diffeomorphism on a domain from the knowledge of a small number of point displacements. This will generate an interesting new version of interpolating splines, which guarantees the creation of non-ambiguous matching. The next example will deal with images, considered as square integrable functions on a domain, for which the action will still consist of diffeomorphisms. The computation of geodesics with this method provides a new algorithm for dense image matching. The obtained "Riemannian" structure (which is infinite dimensional) can also be used to generate normal local charts, which yields a deformation compliant way to index deformable templates relatively to a prototype (template).