PDE-LDDMM meets NODEs: Introducing neural ordinary differential equation solvers in PDE-constrained Large Deformation Diffeomorphic Metric Mapping

Non-rigid image registration is a crucial task in various medical applications, allowing the alignment of images with complex spatial or temporal variations. This paper introduces NODEO-LDDMM and NODEO-PDE-LDDMM, two innovative deep-learning-based approaches that bridge the gap between Large Deformation Diffeomorphic Metric Mapping (LDDMM) and neural ordinary differential equations (NODEs). LDDMM and PDE-LDDMM offer mathematically well-established formulations for diffeomorphic registration, while NODEs provide the flexibility of deep-learning in the solution of the ODEs involved in both methods. Both NODEO-LDDMM and NODEO-PDE-LDDMM include the strengths of deep-learning into LDDMM, enabling a robust optimization with a good balance between accuracy and transformation smoothness in their solutions. Our proposed methods reached or outperformed their traditional counterparts and the nearly diffeomorphic deep-learning-based approaches selected as benchmarks. This work contributes to advancing non-rigid image registration techniques, with a methodology suited to overcome some of the limitations of deep-learning in medical image registration.