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

A Lagrangian Gauss-Newton-Krylov Solver for Mass- and Intensity-Preserving Diffeomorphic Image Registration