Parallel transport in shape analysis: a scalable numerical scheme
This work addresses a domain-specific problem in shape analysis for researchers dealing with high-dimensional manifold data, such as in medical imaging, and is incremental as it adapts an existing scheme.
The paper tackles the computational complexity of analyzing manifold-valued data by adapting a numerical scheme for parallel transport to finite-dimensional manifolds of diffeomorphisms, demonstrating its behavior on high-dimensional manifolds and applying it to predict brain structure progression.
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.