Parallel transport in shape analysis : a scalable numerical scheme

Maxime Louis (1, 2) , Alexandre Bône (1, 2) , Benjamin Charlier (1, 3) , Stanley Durrleman (1, 2)

1 ARAMIS - Algorithms, models and methods for images and signals of the human brain = Algorithmes, modèles et méthodes pour les images et les signaux du cerveau humain [ICM Paris]
2 ICM - Institut du Cerveau et de la Moëlle Epinière = Brain and Spine Institute
3 IMAG - Institut Montpelliérain Alexander Grothendieck

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.

Domaines: Vision par ordinateur et reconnaissance de formes [cs.CV]; Géométrie différentielle [math.DG]; Analyse numérique [math.NA]

Maxime Louis, Alexandre Bône, Benjamin Charlier, Stanley Durrleman. Parallel transport in shape analysis : a scalable numerical scheme. Geometric Science of Information, Nov 2017, Paris, France. ⟨hal-01565478⟩