Learning distributions of shape trajectories from longitudinal datasets:
a hierarchical model on a manifold of diffeomorphisms
Abstract
We propose a method to learn a distribution of shape trajectories from longitudinal data, i.e. the collection of individual objects repeatedly observed at multiple time-points.
The method allows to compute an average spatiotemporal
trajectory of shape changes at the group level, and the individual variations of this trajectory both in terms of geometry
and time dynamics. First, we formulate a non-linear mixedeffects statistical model as the combination of a generic statistical model for manifold-valued longitudinal data, a deformation model defining shape trajectories via the action
of a finite-dimensional set of diffeomorphisms with a manifold structure, and an efficient numerical scheme to compute parallel transport on this manifold. Second, we introduce a MCMC-SAEM algorithm with a specific approach
to shape sampling, an adaptive scheme for proposal variances, and a log-likelihood tempering strategy to estimate
our model. Third, we validate our algorithm on 2D simulated data, and then estimate a scenario of alteration of
the shape of the hippocampus 3D brain structure during the
course of Alzheimer’s disease. The method shows for instance that hippocampal atrophy progresses more quickly
in female subjects, and occurs earlier in APOE4 mutation
carriers. We finally illustrate the potential of our method for
classifying pathological trajectories versus normal ageing