Combining the band-limited parameterization and Semi-Lagrangian Runge--Kutta integration for efficient PDE-constrained LDDMM

TL;DR

This paper introduces three variants of band-limited PDE-constrained LDDMM that combine band-limited parameterization with Semi-Lagrangian Runge--Kutta integration, significantly improving computational efficiency while maintaining high accuracy.

Contribution

It proposes a novel combination of band-limited vector fields and Semi-Lagrangian integration in PDE-constrained LDDMM, enhancing efficiency without sacrificing accuracy.

Findings

All variants show increased computational efficiency.

The deformation state equation variant performs best overall.

The methods maintain high accuracy across evaluations.

Abstract

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.