Riemannian Geometric Statistics in Medical Image Analysis

2

Statistics on manifolds

Tom Fletcher    University of Virginia, Departments of Electrical & Computer Engineering and Computer Science, Charlottesville, VA United States

Abstract

Statistical analysis of data on a Riemannian manifold extends fundamental concepts from multivariate statistical analysis in vector spaces by using the metric structure. The first example of generalizing a classical statistic to the manifold setting is the Fréchet mean, which minimizes the sum-of-squared geodesic distances to the data. Such a geometric least-squares principle also leads to extensions of principal components analysis and regression on manifolds. From these geometric concepts, we extend to a probabilistic viewpoint by defining normal distributions on Riemannian ...