Clustering of longitudinal shape data sets using mixture of separate or branching trajectories

Several methods have been proposed recently to learn spatiotemporal models of shape progression from repeated observations of several subjects over time, i.e. a longitudinal data set. These methods summarize the population by a single common trajectory in a supervised manner. In this paper, we propose to extend such approaches to an unsupervised setting where a longitudinal data set is automatically clustered in different classes without labels. Our method learns for each cluster an average shape trajectory (or representative curve) and its variance in space and time. Representative trajectories are built as the combination of pieces of curves. This mixture model is flexible enough to handle independent trajectories for each cluster as well as fork and merge scenarios. The estimation of such non linear mixture models in high dimension is known to be difficult because of the trapping states effect that hampers the optimisation of cluster assignments during training. We address this issue by using a tempered version of the stochastic EM algorithm. Finally, we apply our algorithm on synthetic data to validate that a tempered scheme achieve better convergence. We show then how the method can be used to test different scenarios of hippocampus atrophy in ageing by using an heteregenous population of normal ageing individuals and mild cognitive impaired subjects.