  The class of non-rigid registration methods proposed in the framework of
PDE-constrained Large Deformation Diffeomorphic Metric Mapping is a
particularly interesting family of physically meaningful diffeomorphic
registration methods. PDE-constrained LDDMM methods are formulated as
constrained variational problems, where the different physical models are
imposed using the associated partial differential equations as hard
constraints. Inexact Newton-Krylov optimization has shown an excellent
numerical accuracy and an extraordinarily fast convergence rate in this
framework. However, the Galerkin representation of the non-stationary velocity
fields does not provide proper geodesic paths. In a previous work, we proposed
a method for PDE-constrained LDDMM parameterized in the space of initial
velocity fields under the EPDiff equation. The proposed method provided
geodesics in the framework of PDE-constrained LDDMM, and it showed performance
competitive to benchmark PDE-constrained LDDMM and EPDiff-LDDMM methods.
However, the major drawback of this method was the large memory load inherent
to PDE-constrained LDDMM methods and the increased computational time with
respect to the benchmark methods. In this work we optimize the computational
complexity of the method using the band-limited vector field parameterization
closing the loop with our previous works.
