Symmetry-Preserving Statistical Models Improve Longitudinal Analysis of Skeletal Configurations with Data from the Osteoarthritis Initiative
Many problems in medical imaging involve the statistical analysis of continuous transformations, such as longitudinal anatomical shape changes and articulated skeletal motion. These transformations are naturally modeled on Lie groups. Statistical methods defined on such spaces should respect these symmetries to ensure invariance to arbitrary choices of coordinate systems and reference frames. In this work, we study geodesic regression on Lie groups as a framework for modeling longitudinal imaging data. While geodesic regression is traditionally formulated using Riemannian metrics, Lie groups generally admit no metric that is fully compatible with their group structure, leading to estimators that break symmetry and may yield unstable results. We therefore propose a non-metric estimator that is equivariant under both left and right group actions. For its computation, we introduce an efficient fixed-point algorithm. The algorithm requires only simple differential expressions, which can be evaluated using automatic differentiation. We validate the approach on synthetic data and apply it to an open-access clinical imaging dataset of longitudinal knee joint configurations acquired to study osteoarthritis progression. The proposed method accurately captures disease-related changes in joint configuration over time. Statistical hypothesis tests based on the estimated geodesic trajectories reveal significant differences between osteoarthritis subgroups and healthy controls. State-of-the-art Riemannian approaches exhibit instability, whereas the proposed bi-invariant method yields robust and reproducible statistical results. These findings demonstrate the advantages of symmetry-preserving statistical modeling for longitudinal medical imaging studies.
