  We propose regularization schemes for deformable registration and efficient
algorithms for their numerical approximation. We treat image registration as a
variational optimal control problem. The deformation map is parametrized by its
velocity. Tikhonov regularization ensures well-posedness. Our scheme augments
standard smoothness regularization operators based on $H^1$- and
$H^2$-seminorms with a constraint on the divergence of the velocity field,
which resembles variational formulations for Stokes incompressible flows. In
our formulation, we invert for a stationary velocity field and a mass source
map. This allows us to explicitly control the compressibility of the
deformation map and by that the determinant of the deformation gradient. We
also introduce a new regularization scheme that allows us to control shear. We
use a globalized, preconditioned, matrix-free, reduced space
(Gauss--)Newton--Krylov scheme for numerical optimization. We exploit variable
elimination techniques to reduce the number of unknowns of our system; we only
iterate on the reduced space of the velocity field. Our current implementation
is limited to the two-dimensional case. The numerical experiments demonstrate
that we can control the determinant of the deformation gradient without
compromising registration quality. This additional control allows us to avoid
oversmoothing of the deformation map. We also demonstrate that we can promote
or penalize shear while controlling the determinant of the deformation
gradient.
