  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 mixed-effects 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.
