Statistics Computing on Manifolds: from Riemannian Geometry to Computational Anatomy

Xavier Pennec
Institut National de Recherche en Informatique Automatique (INRIA)
Projet Epidaure

Computational anatomy aims at modeling the biological variability of the human anatomy.
To reach this goal, the method is to identify anatomically representative geometric features (points, tensors, curves, surfaces, volume transformations), and to describe and compare their statistical distribution in different populations. Unfortunately, geometric features often belong to manifolds that are not vector spaces. Based on a Riemannian manifold structure, we will detail how one can develop a consistent framework for statistical computing on manifolds, starting with the notions of mean value and covariance matrix of a random element, normal law, Mahalanobis distance and test. Then, we will extend the Riemannian computing framework to PDEs for smoothing and interpolation of fields of geometric elements with the example of positive define symmetric matrices (tensors). We show that the choice of a convenient Riemannian metric allows to generalize consistently to tensor fields many important geometric data processing algorithms such as interpolation, filtering, diffusion and restoration of missing data. This framework will be exemplified with the modeling of the brain variability from a dataset of lines on the cerebral cortex. The resulting dense 3D variability map can be seen as the diagonal elements of the Green's function of the Brain accross subjects. This modeling can be extended with non-diagonal element by computing significantly correlated regions in the brain. Finally, we will discuss some of the methods that have been recently introduced to compute statistics on diffeomorphisms.

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