Trajectories from Distribution-Valued Functional Curves: A Unified Wasserstein Framework

Temporal changes in medical images are often evaluated along a parametrized function that represents a structure of interest (e.g. white matter tracts). By attributing samples along these functions with distributions of image properties in the local neighborhood, we create distribution-valued signatures for these functions. We propose a novel, comprehensive framework which models their temporal evolution trajectories. This is achieved under the unifying scheme of Wasserstein distance metric. The regression problem is formulated as a constrained optimization problem and solved using an alternating projection algorithm. The solution simultaneously preserves the functional characteristics of the curve, models the temporal change in distribution profiles and forces the estimated distributions to be valid. Hypothesis testing is applied in two ways using Wasserstein based test statistics. Validation is presented on synthetic data. Estimation of a population trajectory is shown using diffusion properties along DTI tracts from a healthy population of infants. Detection of delayed growth is shown using a case study.