Parallel transport in shape analysis: a scalable numerical scheme

TL;DR

This paper introduces a scalable numerical scheme for parallel transport in shape analysis, enabling efficient computations on high-dimensional Riemannian manifolds, with applications to brain structure progression prediction.

Contribution

It adapts a recent generic numerical scheme for parallel transport to finite-dimensional diffeomorphism manifolds, addressing computational challenges in high-dimensional shape analysis.

Findings

The scheme performs well on high-dimensional manifolds.

It effectively predicts brain structure progression.

The method is scalable and numerically stable.

Abstract

The analysis of manifold-valued data requires efficient tools from Riemannian geometry to cope with the computational complexity at stake. This complexity arises from the always-increasing dimension of the data, and the absence of closed-form expressions to basic operations such as the Riemannian logarithm. In this paper, we adapt a generic numerical scheme recently introduced for computing parallel transport along geodesics in a Riemannian manifold to finite-dimensional manifolds of diffeomorphisms. We provide a qualitative and quantitative analysis of its behavior on high-dimensional manifolds, and investigate an application with the prediction of brain structures progression.