  The family of PDE-constrained LDDMM methods is emerging as a particularly
interesting approach for physically meaningful diffeomorphic transformations.
The original combination of Gauss--Newton--Krylov optimization and Runge--Kutta
integration, shows excellent numerical accuracy and fast convergence rate.
However, its most significant limitation is the huge computational complexity,
hindering its extensive use in Computational Anatomy applied studies. This
limitation has been treated independently by the problem formulation in the
space of band-limited vector fields and Semi-Lagrangian integration. The
purpose of this work is to combine both in three variants of band-limited
PDE-constrained LDDMM for further increasing their computational efficiency.
The accuracy of the resulting methods is evaluated extensively. For all the
variants, the proposed combined approach shows a significant increment of the
computational efficiency. In addition, the variant based on the deformation
state equation is positioned consistently as the best performing method across
all the evaluation frameworks in terms of accuracy and efficiency.
