## Geodesic Motions and Vector Fields on Anatomical Manifolds

#### Alain TrouvéEcole Normale Supérieure, FranceL.A.G.A

The anatomical orbits or deformable templates are made into a metric space with a
metric distance between elements by constructing curves through the space of
diffeomorphisms connecting them; the length of the curve becomes the basis for the
construction, the metric distance corresponding to the geodesic shortest length curves.
This gives rise to a natural variational problem describing the geodesic flows between
elements in the orbit, with the solution of the associated Euler-Lagrange equations giving
the optimal flow of diffeomorphisms and thus the metric between the shapes. The
obtained setting shares several similarities with the mechanics of perfect fluids, for which
the Euler-Lagrange equation has been derived by Arnold (Equation 1 of Arnold 1992) for
the group of divergence-free volume-preserving diffeomorphisms. As well these results
become another example of the general Euler-Poincaré principle of Marsden & Ratiu
(1994) and Holm et al (1998) but applied to the infinite dimensional setting.

Such a point of view will link our geodesic formulation to a "conservation of momentum
law" in Lagrangian coordinates providing a powerful method for studying and modeling
diffeomorphic evolution of shape. It will imply that the momentum of the diffeomorphic
flow at any place along the geodesic can be generated from the momentum at the
origin, thus providing the vehicle for "geodesic generation via shooting".

V.I. Arnold. Topological methods in hydrodynamics Ann. Rev. Fluid. Mech. 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. Adv. in Math., 137:1--81, 1998.

Back to Graduate Summer School: Mathematics in Brain Imaging