Variational methods for image registration basically involve a regularizer to ensure that the resulting well-posed problem admits a solution. Different choices of regularizers lead to different deformations. On one hand, the conventional regularizers, such as the elastic, diffusion and curvature regularizers, are able to generate globally smooth deformations and generally useful for many applications. On the other hand, these regularizers become poor in some applications where discontinuities or steep gradients in the deformations are required. As is well-known, the total (TV) variation regularizer is more appropriate to preserve discontinuities of the deformations. However, it is difficult in developing an efficient numerical method to ensure that numerical solutions satisfy this requirement because of the non-differentiability and non-linearity of the TV regularizer. In this work we focus on computational challenges arising in approximately solving TV-based image registration model. Motivated by many efficient numerical algorithms in image restoration, we propose to use augmented Lagrangian method (ALM). At each iteration, the computation of our ALM requires to solve two subproblems. On one hand for the first subproblem, it is impossible to obtain exact solution. On the other hand for the second subproblem, it has a closed-form solution. To this end, we propose an efficient nonlinear multigrid (NMG) method to obtain an approximate solution to the first subproblem. Numerical results on real medical images not only confirm that our proposed ALM is more computationally efficient than some existing methods, but also that the proposed ALM delivers the accurate registration results with the desired property of the constructed deformations in a reasonable number of iterations.
A partial parallel splitting augmented Lagrangian method for solving constrained matrix optimization problems
