Band-Limited Stokes Large Deformation Diffeomorphic Metric Mapping

Insights into traditional Large Deformation Diffeomorphic Metric Mapping and unsupervised deep-learning for diffeomorphic registration and their evaluation

This paper explores the connections between traditional Large Deformation Diffeomorphic Metric Mapping methods and unsupervised deep-learning approaches for non-rigid registration, particularly emphasizing diffeomorphic registration. The study provides useful insights and establishes connections between the methods, thereby facilitating a profound understanding of the methodological landscape. The methods considered in our study are extensively evaluated in T1w MRI images using traditional NIREP and Learn2Reg OASIS evaluation protocols with a focus on fairness, to establish equitable benchmarks and facilitate informed comparisons. Through a comprehensive analysis of the results, we address key questions, including the intricate relationship between accuracy and transformation quality in performance, the disentanglement of the influence of registration ingredients on performance, and the determination of benchmark methods and baselines. We offer valuable insights into the strengths and limitations of both traditional and deep-learning methods, shedding light on their comparative performance and guiding future advancements in the field.

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

This work proposes a unifying framework for extending PDE-constrained Large Deformation Diffeomorphic Metric Mapping (PDE-LDDMM) with the sum of squared differences (SSD) to PDE-LDDMM with different image similarity metrics. We focused on the two best-performing variants of PDE-LDDMM with the spatial and band-limited parameterizations of diffeomorphisms. We derived the equations for gradient-descent and Gauss-Newton-Krylov (GNK) optimization with Normalized Cross-Correlation (NCC), its local version (lNCC), Normalized Gradient Fields (NGFs), and Mutual Information (MI). PDE-LDDMM with GNK was successfully implemented for NCC and lNCC, substantially improving the registration results of SSD. For these metrics, GNK optimization outperformed gradient-descent. However, for NGFs, GNK optimization was not able to overpass the performance of gradient-descent. For MI, GNK optimization involved the product of huge dense matrices, requesting an unaffordable memory load. The extensive evaluation reported the band-limited version of PDE-LDDMM based on the deformation state equation with NCC and lNCC image similarities among the best performing PDE-LDDMM methods. In comparison with benchmark deep learning-based methods, our proposal reached or surpassed the accuracy of the best-performing models. In NIREP16, several configurations of PDE-LDDMM outperformed ANTS-lNCC, the best benchmark method. Although NGFs and MI usually underperformed the other metrics in our evaluation, these metrics showed potentially competitive results in a multimodal deformable experiment. We believe that our proposed image similarity extension over PDE-LDDMM will promote the use of physically meaningful diffeomorphisms in a wide variety of clinical applications depending on deformable image registration.