The Euler-Poincaré theory of metamorphosis

Abstract

In the pattern matching approach to imaging science, the process of “metamorphosis” is template matching with dynamical templates (Trouvé and Younes, Found. Comp. Math. , 2005). Here, we recast the metamorphosis equations of that paper into the Euler-Poincaré variational framework of Holm, Marsden, and Ratiu, Adv. in Math. , 1998 and show that the metamorphosis equations contain the equations for a perfect complex fluid (Holm, Springer , 2002). This result connects the ideas underlying the process of metamorphosis in image matching to the physical concept of an order parameter in the theory of complex fluids. After developing the general theory, we reinterpret various examples, including point set, image and density metamorphosis. We finally discuss the issue of matching measures with metamorphosis, for which we provide existence theorems for the initial and boundary value problems.