Nonparametric Aggregation of Geodesic Trends for Longitudinal Data Analysis

We propose a technique for analyzing longitudinal imaging data that models individual changes with diffeomorphic geodesic regression and aggregates these geodesics into a nonparametric group average trend. Our model is specifically tailored to the unbalanced and sparse characteristics of longitudinal imaging studies. That is, each individual has few data points measured over a short period of time, while the study population as a whole spans a wide age range. We use geodesic regression to estimate individual trends, which is an appropriate model for capturing shape changes over a short time window, as is typically found within an individual. Geodesics are also adept at handling the low sample sizes found within individuals, and can model the change between as few as two timepoints. However, geodesics are limited for modeling longer-term trends, where constant velocity may not be appropriate. Therefore, we develop a novel nonparametric regression to aggregate individual trends into an average group trend. We demonstrate the power of our method to capture non-geodesic group trends on hippocampal volume (real-valued data) and diffeomorphic registration of full 3D MRI from the longitudinal OASIS data.