Geometric statistics in computational anatomy: old & new

The data of computational anatomy are usually organs shapes extracted from medical images. In order to analyze them independently of their parametrisation, one have to deal with equivalence classes of sets of points, curves, surfaces or images under the action of a reparametrisation group. In neuroimaging, connectomes extracted from functional MRI are encoded by the correlation between signals at difference parcels of the brain, that is, the quotient of the SPD covariance matrices by diagonal rescalings, which is once again a quotient space. But quotient spaces are almost always non-linear spaces, while statistics where essentially developed in a Euclidean setting. Thus, there is a need for redefining a consistent statistical framework for objects living in manifolds and Lie groups, a field which is now called geometric statistics. The objective of this talk is to give an overview of geometric statistics methods in Riemannian manifolds and some of its recent developments. The talk is motivated and illustrated by applications in medical image analysis, such as the regression of simple and efficient models of the atrophy of the brain in Alzheimer's disease using the parallel transport of image deformations and the recently proposed metrics on spaces of correlation matrices with applications in connectomics. We will also show that classical statistical tools like Principal Component Analysis (PCA) most often suffer in practice from a stability and interpretability problem (the curse of isotropy) and should be replaced by Principal Subspace analysis, a relaxation of PCA based on flag spaces (sequences of nested subspaces generalizing Grassmannians).