A Comparative Study of Different Variants of Newton-Krylov PDE-Constrained Stokes-LDDMM Parameterized in the Space of Band-Limited Vector Fields

Monica Hernandez

Abstract: PDE-constrained Stokes Large Deformation Diffeomorphic Metric Mapping (LDDMM) is a particularly interesting framework of physically meaningful diffeomorphic registration methods. PDE-constrained Stokes-LDDMM is formulated as a constrained variational problem, where the different physical models can be introduced using the associated partial differential equations as hard constraints. Newton--Krylov optimization has shown an excellent numerical accuracy and an extraordinarily fast convergence rate in this framework. However, the most significant limitation of PDE-constrained Stokes LDDMM is the huge computational complexity, which hinders the extensive use in Computational Anatomy applied studies. In previous work, we proposed a time-efficient approximation of PDE-constrained Stokes-LDDMM formulating the constrained variational problem in the space of band-limited vector fields and performing the computations in the GPU. The parameterization in the space of band-limited vector fields dramatically reduced the computation time. However, the method still showed a considerable memory load, and the performance was low for the smallest and most interesting domain sizes. In this work, we propose two novel variants of Newton--Krylov PDE-constrained Stokes-LDDMM parameterized in the space of band-limited vector fields. The variants are better suited for the band-limited vector field parameterization, yielding efficient methods in terms of time and memory. We have performed a comparative study of the performance of our proposed methods with our previous contribution and the state-of-the-art methods most related to our work. The results have shown that some configurations of the proposed variants outperform the state-of-the-art methods in terms of accuracy and computational complexity. The proposed variants can be regarded as a potentially useful contribution to the set of physically meaningful LDDMM registration methods.