Parallel Transport of Surface Deformations from Pole Ladder to Symmetrical Extension

Abstract

Cardiac motion contains information underlying disease development, and complements the anatomical information extracted for each subject. However, normalization of temporal trajectories is necessary due to anatomical differences between subjects. In this study, we encode inter-subject shape variations and temporal deformations in a common space of diffeomorphic registration. They are parameterized by stationary velocity fields. Previous normalization algorithms applied in medical imaging were first order approximations of parallel transport. In contrast, pole ladder was recently shown to be a third order scheme in general affine connection spaces and exact in one step in affine symmetric spaces. We further improve this procedure with a more symmetric mapping scheme, which relies on geodesic symmetries around mid-points. We apply the method to analyze cardiac motion among pulmonary hypertension populations. Evaluation is performed on a 4D cardiac database, with meshes of the right-ventricle obtained by commercial speckle-tracking from echo-cardiogram. We assess the stability of the algorithms by computing their numerical inverse error. Our method turns out to be very accurate and efficient in terms of compactness for subspace representation.