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.

我们提出了一种从纵向数据（即在多个时间点反复观察到的个体对象的集合）中学习形状轨迹分布的方法，并介绍了具体的MCMC-SAEM算法，用于评估我们的模型，应用于阿尔茨海默病中的海马3D脑结构形状变化情景的估计，以及分类病理轨迹与正常衰老的潜力的演示。

从纵向数据集中学习形状轨迹的分布：基于微分同胚流形的层次模型