Riemannian Geometric Statistics in Medical Image Analysis

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Introduction to differential and Riemannian geometry

Stefan Sommer^a; Tom Fletcher^b; Xavier Pennec^c    aUniversity of Copenhagen, Department of Computer Science, Copenhagen, Denmark^bUniversity of Virginia, Departments of Electrical & Computer Engineering and Computer Science, Charlottesville, VA, United States^cUniversité Côte d'Azur and Inria, Epione team, Sophia Antipolis, France

Abstract

This chapter introduces the basic concepts of differential geometry: Manifolds, charts, curves, their derivatives, and tangent spaces. The addition of a Riemannian metric enables length and angle measurements on tangent spaces giving rise to the notions of curve length, geodesics, and thereby the basic constructs for statistical analysis of manifold-valued data. ...