Parallel transport, a central tool in geometric statistics for computational anatomy: Application to cardiac motion modeling

Abstract

Transporting the statistical knowledge regressed in the neighborhood of a point to a different but related place (transfer learning) is important for many applications. In medical imaging, cardiac motion modeling and structural brain changes are two such examples: for a group-wise statistical analysis, subject-specific longitudinal deformations need to be transported in a common template anatomy. In geometric statistics, the natural (parallel) transport method is defined by the integration of a Riemannian connection which specifies how tangent vectors are compared at neighboring points. In this process, the numerical accuracy of the transport method is critical. Discrete methods based on iterated geodesic parallelograms inspired by Schild's ladder were shown to be very efficient and apparently stable in practice. In this chapter, we show that ladder methods are actually second-order schemes, even with numerically approximated geodesics. We also propose a new original algorithm to implement these methods in the context of the large deformation diffeomorphic metric mapping (LDDMM) framework that endows the space of diffeomorphisms with a right-invariant RKHS metric. When applied to the motion modeling of the cardiac right ventricle under pressure or volume overload, the method however exhibits unexpected effects in the presence of very large volume differences between subjects. We first investigate an intuitive rescaling of the modulus after parallel transport to preserve the ejection fraction. The surprisingly simple scaling/volume relationship that we obtain suggests to decouples the volume change from the deformation directly within the LDDMM metric. The parallel transport of cardiac trajectories with this new metric now reveals statistical insights into the dynamics of each disease. This example shows that parallel transport could become a tool of choice for data-driven metric optimization.