Geodesic Motions and Vector Fields on Anatomical Manifolds
Alain Trouvé
Ecole Normale Supérieure, France
L.A.G.A
Anatomical orbits, or deformable templates, are endowed with a metric structure by constructing curves in the space of diffeomorphisms that connect them. The length of such a curve provides the basis for defining the metric distance, with the geodesic—i.e., the shortest path—yielding the distance between elements. This leads to a natural variational problem describing geodesic flows between elements in the orbit. Solving the associated Euler–Lagrange equations determines the optimal flow of diffeomorphisms and thus the metric between shapes.
This framework shares notable similarities with the mechanics of perfect fluids. In that context, Arnold (1992) derived the Euler–Lagrange equations for the group of divergence-free, volume-preserving diffeomorphisms. More broadly, these results constitute another example of the Euler–Poincaré principle developed by Marsden and Ratiu (1994) and by Holm, Marsden, and Ratiu (1998), extended to an infinite-dimensional setting.
This perspective connects the geodesic formulation to a conservation-of-momentum law in Lagrangian coordinates, providing a powerful method for studying and modeling diffeomorphic shape evolution. In particular, the momentum of the diffeomorphic flow at any point along the geodesic can be generated from the initial momentum, enabling geodesic generation via shooting.
References:
V. I. Arnold. “Topological methods in hydrodynamics.” Annual Review of Fluid Mechanics, 24:145–166, 1992.
J. E. Marsden and T. S. Ratiu. Introduction to Mechanics and Symmetry. Springer, 1994.
D. D. Holm, J. E. Marsden, and T. S. Ratiu. “The Euler–Poincaré equations and semidirect products with applications to continuum theories.” Advances in Mathematics, 137:1–81, 1998.
