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\begin{abstract}
Spatial transformations that capture population-level morphological statistics are critical for medical image analysis. Commonly used smoothness regularizers for image registration fail to integrate population statistics, leading to anatomically inconsistent transformations.
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Inverse consistency regularizers promote geometric consistency but lack population morphometrics integration. Regularizers that constrain deformation to low-dimensional manifold methods address this. However, they prioritize reconstruction over interpretability and neglect diffeomorphic properties, such as group composition and inverse consistency. 
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We introduce \model, a Log-Euclidean regularization framework for population-aware unsupervised image registration. \model~learns population morphometrics from spatial transformations to guide and regularize registration networks, ensuring anatomically plausible deformations. It features a bottleneck autoencoder that computes the principal logarithm of deformation fields via iterative square-root predictions. It creates a linearized latent space that respects diffeomorphic properties and enforces inverse consistency.
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By integrating a registration network with a diffeomorphic autoencoder, \model~produces smooth, meaningful deformation fields. The framework offers two main contributions: (1) a data-driven regularization strategy that incorporates population-level anatomical statistics to enhance transformation validity and (2) a linearized latent space that enables compact and interpretable deformation fields for efficient population morphometrics analysis.
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We validate \model~across two families of deep learning-based registration networks, demonstrating its ability to produce anatomically accurate, computationally efficient, and statistically meaningful transformations on the OASIS-1 brain imaging dataset. \href{https://github.com/iyerkrithika21/MORPH_LER}{https://github.com/iyerkrithika21/MORPH\_LER}
\end{abstract}




\begin{keywords}
deformable image registration, manifold statistics, non-
rigid, diffeomorphisms, shape population statistics, log-
euclidean statistics
\end{keywords}