MULTISCALE APPROACH FOR TWO-DIMENSIONAL DIFFEOMORPHIC IMAGE REGISTRATION*

In a beautiful paper, Modin, Nachman, and Rondi [Adv. Math., 346 (2019), pp. 1009-1066] introduced a hierarchical image registration model based on the large deformation diffeomorphic metric mapping (LDDMM) framework. Unfortunately, no numerical tests are performed to show the efficiency of this multiscale approach. The LDDMM image registration framework is essentially a variational problem with differential equation constraints and the structure of the cost functional is very complex. Therefore, it's necessary and meaningful to introduce some other analogous multiscale approaches with a much simpler cost functional. Motivated by the work of Modin, Nachman, and Rondi, we construct a multiscale image registration approach for the two-dimensional diffeomorphic image registration model in [H. Han and Z. Wang, SIAM J. Imaging Sci., 13 (2020), pp. 1240-1271]. This approach achieves a smooth minimizer for the cost functional without regularization. This result is completely different from most published models which only achieve minimizers of the cost functional with some regularization. The existence of solutions for the multiscale approach and the convergence of the multiscale approach are proved. In addition, a multigrid based multi scale diffeomorphic image registration algorithm is presented. Moreover, numerical tests are also performed to show that the proposed multiscale approach achieves a satisfactory image registration result without mesh folding.