An inexact Newton-Krylov algorithm for constrained diffeomorphic image registration

We propose numerical algorithms for solving large deformation diffeomorphic image registration problems. We formulate the non-rigid image registration problem as a problem of optimal control. This leads to a PDE constrained optimization problem. The PDE constraint consists, in its simplest form, of a hyperbolic transport equation for the evolution of the image intensity. The control variable is the velocity field. Tikhonov regularization ensures well-posedness. We consider standard smoothness regularization based on $H^1$- or $H^2$-seminorms. We augment this regularization scheme with a constraint on the divergence of the velocity field (Stokes regularization scheme). We use a Fourier pseudo-spectral discretization in space and a Chebyshev pseudo-spectral discretization in time. The latter allows us to reduce the number of unknowns and enables the time-adaptive inversion for non-stationary velocity fields. We use a preconditioned, globalized, matrix-free, inexact Newton-Krylov method for numerical optimization. A parameter continuation is designed to estimate an optimal regularization parameter. Regularity is ensured by controlling the geometric properties of the deformation field. Overall, we arrive at a black-box solver that exploits computational tools that are precisely tailored for solving the optimality system. We study spectral properties of the Hessian, grid convergence, numerical accuracy, computational efficiency, and deformation regularity of our scheme. We compare the designed Newton-Krylov methods with a globalized Picard method (preconditioned gradient descent). The reported results demonstrate excellent numerical accuracy, guaranteed local deformation regularity, and computational efficiency with an optional control on local mass conservation. The Newton-Krylov methods clearly outperform the Picard method if high accuracy of the inversion is required.