Image registration is a core task in computational anatomy that establishes correspondences between images. Invertible deformable registration, which computes a deformation field and handles complex, non-linear transformation, is essential for tracking anatomical variations, especially in neuroimaging applications where inter-subject differences and longitudinal changes are key. Analyzing the deformation fields is challenging due to their non-linearity, limiting statistical analysis. However, traditional approaches for analyzing deformation fields are computationally expensive, sensitive to initialization, and prone to numerical errors, especially when the deformation is far from the identity. To address these limitations, we propose the Log-Euclidean Diffeomorphic Autoencoder (LEDA), an innovative framework designed to compute the principal logarithm of deformation fields by efficiently predicting consecutive square roots. LEDA operates within a linearized latent space that adheres to the diffeomorphisms group action laws, enhancing our model's robustness and applicability. We also introduce a loss function to enforce inverse consistency, ensuring accurate latent representations of deformation fields. Extensive experiments with the OASIS-1 dataset demonstrate the effectiveness of LEDA in accurately modeling and analyzing complex non-linear deformations while maintaining inverse consistency. Additionally, we evaluate its ability to capture and incorporate clinical variables, enhancing its relevance for clinical applications.

翻译：图像配准是计算解剖学中的核心任务，旨在建立图像间的对应关系。可逆形变配准通过计算形变场并处理复杂的非线性变换，对于追踪解剖结构变化至关重要，尤其是在关注个体间差异与纵向变化的神经影像应用中。由于形变场的非线性特性，其统计分析面临挑战。然而，传统的形变场分析方法计算成本高昂，对初始化敏感，且易产生数值误差，尤其在形变远离恒等变换时更为明显。为克服这些局限，我们提出Log-Euclidean微分同胚自编码器（LEDA），该创新框架通过高效预测连续平方根来计算形变场的主对数。LEDA在遵循微分同胚群作用规律的线性化潜在空间中运行，从而提升了模型的鲁棒性与适用性。我们还引入了一种强制逆一致性的损失函数，以确保形变场潜在表示的准确性。基于OASIS-1数据集的大量实验表明，LEDA能够精确建模和分析复杂非线性形变，同时保持逆一致性。此外，我们评估了其捕获并整合临床变量的能力，进一步增强了其在临床应用中的相关性。