Understanding Trade-Offs in Continuous Neural Representations for Diffeomorphic Image Registration: A Comparative Study of Implicit Neural Representations and Neural Ordinary Differential Equations

Non-rigid image registration is a fundamental problem in medical imaging and a representative example of continuous transformation modeling in image processing. Diffeomorphic registration methods, such as Large Deformation Diffeomorphic Metric Mapping (LDDMM) and its PDE-constrained variants (PDE-LDDMM), provide mathematically grounded formulations with strong geometric guarantees for transformation quality. However, existing approaches face persistent trade-offs between numerical stability, accuracy, and computational efficiency. Recent work has explored implicit neural representations (INRs) and neural ordinary differential equations (NODEs) as flexible neural representations for modeling continuous transformations. Despite their increasing adoption, their practical behavior and limitations in diffeomorphic registration remain insufficiently understood. In this paper, we present a unified formulation of INR- and NODE-based registration methods within LDDMM and PDE-LDDMM, enabling a systematic and controlled comparison across architectures, sampling strategies, and numerical solvers. Our analysis reveals fundamental trade-offs between these approaches. In particular, we show that MLP-based INR formulations introduce significant computational overhead and rely on sampling strategies that can degrade smoothness and lead to the increased occurrence of non-diffeomorphic transformations at higher resolutions. Moreover, these approximations do not fully alleviate the computational cost, with some variants exceeding the costs of expensive classical optimization-based methods. In contrast, NODE-based formulations and downsampling strategies consistently provide transformations with more controlled Jacobian extrema while maintaining competitive computational performance. Among the evaluated methods, the original NODE-LDDMM and NODE-PDE-LDDMM formulations achieve the most favorable trade-offs between registration accuracy, geometric consistency, and computational efficiency. These findings provide clear insights into the design of neural representations for continuous transformation modeling, with practical implications for diffeomorphic registration and computational anatomy applications.