Discrete ladders for parallel transport in transformation groups with an affine connection structure

Marco Lorenzi

Xavier Pennec

pp. 243-271

in: Frank Nielsen (ed), Geometric theory of information, Berlin, Springer, 2014

Abstract

The analysis of complex information in medical imaging and in computer vision often requires to represent data in suitable manifolds. In this scenario we are interested in studying the mathematical relationship between objects belonging to the same geometrical space, for instance by calculating trajectories, distances, and statistical modes. An important example can be found in computational anatomy, in which we aim at developing statistical models for the study of the anatomical variability of organs and tissues. In particular, temporal evolutions of anatomies can be modeled by transformations of the space represented by trajectories of diffeomorphisms. Reliable methods for comparing different trajectories are thus required in order to develop population-based models. In this chapter we provide fundamental notions of finite-dimensional Lie Groups and Riemannian geometry which are the basis of the classical continuous parallel transport methods. We introduce then the Schild's ladder, an efficient and simple method proposed in theoretical Physics for the parallel transport of vectors along geodesics paths by iterative construction of infinitesimal geodesics parallelograms on the manifolds. Schild's ladder may be however inefficient for transporting longitudinal deformations from image time series of multiple time points, in which the computation of the geodesic diagonals is required several times. We propose therefore a new parallel transport method based on the Schild's ladder, the "pole ladder", in which the computation of geodesics diagonals is minimized. These theoretical concepts are then contextualized and discussed in the applicative setting of diffeomorphic image registration, and applied to the very practical problem of statistical analysis of the group-wise longitudinal brain changes in Alzheimer's disease.