  The class of non-rigid registration methods proposed in the framework of
PDE-constrained Large Deformation Diffeomorphic Metric Mapping is a
particularly interesting family of physically meaningful diffeomorphic
registration methods. Inexact Newton-Krylov optimization has shown an excellent
numerical accuracy and an extraordinarily fast convergence rate in this
framework. However, the Galerkin representation of the non-stationary velocity
fields does not provide proper geodesic paths. In this work, we propose a
method for PDE-constrained LDDMM parameterized in the space of initial velocity
fields under the EPDiff equation. The derivation of the gradient and the
Hessian-vector products are performed on the final velocity field and
transported backward using the adjoint and the incremental adjoint Jacobi
equations. This way, we avoid the complex dependence on the initial velocity
field in the derivations and the computation of the adjoint equation and its
incremental counterpart. The proposed method provides geodesics in the
framework of PDE-constrained LDDMM, and it shows performance competitive to
benchmark PDE-constrained LDDMM and EPDiff-LDDMM methods.
