  We present an efficient solver for diffeomorphic image registration problems
in the framework of Large Deformations Diffeomorphic Metric Mappings (LDDMM).
We use an optimal control formulation, in which the velocity field of a
hyperbolic PDE needs to be found such that the distance between the final state
of the system (the transformed/transported template image) and the observation
(the reference image) is minimized. Our solver supports both stationary and
non-stationary (i.e., transient or time-dependent) velocity fields. As
transformation models, we consider both the transport equation (assuming
intensities are preserved during the deformation) and the continuity equation
(assuming mass-preservation).
  We consider the reduced form of the optimal control problem and solve the
resulting unconstrained optimization problem using a discretize-then-optimize
approach. A key contribution is the elimination of the PDE constraint using a
Lagrangian hyperbolic PDE solver. Lagrangian methods rely on the concept of
characteristic curves that we approximate here using a fourth-order Runge-Kutta
method. We also present an efficient algorithm for computing the derivatives of
final state of the system with respect to the velocity field. This allows us to
use fast Gauss-Newton based methods. We present quickly converging iterative
linear solvers using spectral preconditioners that render the overall
optimization efficient and scalable. Our method is embedded into the image
registration framework FAIR and, thus, supports the most commonly used
similarity measures and regularization functionals. We demonstrate the
potential of our new approach using several synthetic and real world test
problems with up to 14.7 million degrees of freedom.
