As pointed out by D'arcy Thomspon in his celebrated "Theory of Transformations" (Growth and Forms, 1917), the structure of a collection of shapes can be more easily understood if one considers that "simple" transformations can usually morph one shape into another. Its mathematical translation into the idea of action of groups of transformations on various "deformable" objects has initiated a rich theoretical and computational framework connecting numerous branches of mathematics fueled by applications in medical imaging, computational anatomy and computational medicine. However, there are still challenging issues to address theoretically and computationally the more difficult \emph{pattern theoretic} question of the "understanding" of the relations between a collection of shapes through the use and the selection of a descriptive language for deformations as pioneered by Ulf Grenander. We will discuss in this talk some of these issues as well as some of our attempts in that direction.

This presentation is part of Minisymposium “MS28 - Diffeomorphic Image Registration: Numerics, Applications, and Theory (2 parts)”
organized by: Andreas Mang (Department of Mathematics, University of Houston) , George Biros (Institute for Computational Engineering and Sciences, University of Texas at Austin) .

Authors:

Alain Trouvé (Centre de Mathématiques et Leurs Applications)

Keywords: