Manifold valued statistical models for longitudinal analysis of MRI data

This work presents novel statistical image analysis methods to characterize complex morphological brain changes using MRI data. Specifically, our procedure utilizes the fundamental representations of "longitudinal change" -- voxel-wise Jacobian matrices obtained from image registration. Currently their univariate summaries (for example determinants) are ubiquitously used in neuroimaging studies. Operating directly with representations of Jacobians namely Cauchy deformation tensors, which are elements of an abstract mathematical manifold of symmetric positive definite matrices, yields promising improvements in statistical power in detecting subtle but statistically significant effects. The key technical contributions are computational algorithms for estimating multivariate general linear models with manifold-valued response variables.