Combining the Band-Limited Parameterization and Semi-Lagrangian Runge–Kutta Integration for Efficient PDE-Constrained LDDMM

Abstract

The family of PDE-constrained Large Deformation Diffeomorphic Metric Mapping (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.

Partial Differential Equation-Constrained Diffeomorphic Registration from Sum of Squared Differences to Normalized Cross-Correlation, Normalized Gradient Fields, and Mutual Information: A Unifying Framework

“…Although the values for the lNCC and MI metrics ranged higher than , their performance in the evaluation reported a similar distribution. For lNCC, NGFs, and MI, the correlation between the lowest values and the highest DSC results that are usually seen for SSD in previous works does not hold anymore [ 22 , 33 ].…”

“…In this section, we show the experiments conducted to evaluate the performance of the two PDE-LDDMM variants for the different image similarity metrics. First, we provide an extensive evaluation of our proposed methods in the NIREP16 database, where we have extensively evaluated previous LDDMM and PDE-LDDMM registration methods [ 20 , 22 , 33 , 67 ]. Next, we evaluate our proposed methods in Klein et al databases.…”

“…The BL experiments were performed with band sizes of for BL Variants I and II. This selection was found as optimal for each method in our previous work [ 20 , 22 , 33 ].…”

“…To integrate the PDEs, we used the semi-Lagrangian Runge–Kutta schemes proposed in [ 33 ] for the SSD versions of Variants I and II. The solutions were computed at the Chebyshev–Gauss–Lobatto discretization of the temporal domain .…”

“…They can be considered theoretically but not numerically equivalent formulations of the original one. These variants have been shown to improve the original formulation in terms of registration accuracy and efficiency, in combination with the use of the band-limited parameterization and GPU implementation [ 20 , 22 , 33 ].…”