  Image registration has played an important role in image processing problems,
especially in medical imaging applications. It is well known that when the
deformation is large, many variational models cannot ensure diffeomorphism. In
this paper, we propose a new registration model based on an optimal control
relaxation constraint for large deformation images, which can theoretically
guarantee that the registration mapping is diffeomorphic. We present an
analysis of optimal control relaxation for indirectly seeking the diffeomorphic
transformation of Jacobian determinant equation and its registration
applications, including the construction of diffeomorphic transformation as a
special space. We also provide an existence result for the control increment
optimization problem in the proposed diffeomorphic image registration model
with an optimal control relaxation. Furthermore, a fast iterative scheme based
on the augmented Lagrangian multipliers method (ALMM) is analyzed to solve the
control increment optimization problem, and a convergence analysis is followed.
Finally, a grid unfolding indicator is given, and a robust solving algorithm
for using the deformation correction and backtrack strategy is proposed to
guarantee that the solution is diffeomorphic. Numerical experiments show that
the registration model we proposed can not only get a diffeomorphic mapping
when the deformation is large, but also achieves the state-of-the-art
performance in quantitative evaluations in comparing with other classical
models.
