Geometric dynamics of optimization
Résumé
This paper investigates a family of dynamical systems arising from an evolutionary re-interpretation of certain optimal control and optimization problems. We focus particularly on the application in image registration of the theory of metamorphosis. Metamorphosis is a means of tracking the optimal changes of shape that are necessary for registration of images with various types of data structures, without requiring that the transformations of shape be diffeomorphisms, but penalizing them if they are not. The possibilities of this approach are just beginning to be developed. In particular, metamorphosis and its related variants in the geometric approach to control and optimization can be expected to produce many exciting opportunities for new applications and analysis in geometric dynamics.