Fast linear geodesic shape regression using coupled logdemons registration

Longitudinal brain image series offers the possibility to study individual brain anatomical changes over time. Mathematical models are needed to study such developmental trajectories in detail. In this paper, we present a novel approach to study the individual brain anatomy over time via a linear geodesic shape regression method. In our method, we integrate separate pairwise registrations between the baseline image and the follow-up images into a unified spatial registration plus temporal regression framework. Different from previous geodesic shape regression approaches, which use the LDDMM framework to estimate the brain anatomical change over time, our method is based on the LogDemons method to decrease the computation cost, while maintaining the diffeomorphic property of the deformation over time. Moreover, a temporal regression constraint is explicitly implemented in each optimization iteration to make sure that the entire developmental trajectory can be compactly represented by the baseline image and an optimal stationary velocity field. Our method is mathematically well founded in the Alternating Direction Method of Multipliers (ADMM), which for our image regression application is interpreted in diffeomorphic space instead of Euclidean space. We evaluate our new method on 2D synthetic images and real 3D brain longitudinal image series, and the experiments show promising results in regression accuracy as well as estimated deformations.