Riemannian geometry on shapesand diffeomorphisms: Statistics via actions of the diffeomorphism group

TY - CHAP

T1 - Riemannian geometry on shapesand diffeomorphisms

T2 - Statistics via actions of the diffeomorphism group

AU - Marsland, Stephen

AU - Sommer, Stefan Horst

PY - 2019/9/4

Y1 - 2019/9/4

N2 - Shapes, which can informally be thought of as curves or surfaces marking the outlines of physical items, are examples of objects that are amenable to statistical analysis using Riemannian geometry. The shapes, either described as continuous curves or surfaces or parameterized as a set of landmark points or an image, can be deformed one to another using smooth invertible functions with smooth inverse (diffeomorphisms). The set of diffeomorphisms is an infinite-dimensional manifold and also a group. We can interpret the variability of shapes by the nature of the deformations applied to them, allowing the shapes to be treated as elements of a highly nonlinear Riemannian manifold. In this chapter we introduce the concept of geometric shape analysis, emphasizing the underlying geometric picture and how it relates to a variety of different ways to describe shapes. We describe the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework for shape deformation and show how the shape spaces inherit the geometry of the diffeomorphism group. For the common cases of landmarks and images, we describe the geodesic equations in explicit form. The setting allows us to perform statistical analysis on abstract representations of various types of shapes in one common geometric setting.

AB - Shapes, which can informally be thought of as curves or surfaces marking the outlines of physical items, are examples of objects that are amenable to statistical analysis using Riemannian geometry. The shapes, either described as continuous curves or surfaces or parameterized as a set of landmark points or an image, can be deformed one to another using smooth invertible functions with smooth inverse (diffeomorphisms). The set of diffeomorphisms is an infinite-dimensional manifold and also a group. We can interpret the variability of shapes by the nature of the deformations applied to them, allowing the shapes to be treated as elements of a highly nonlinear Riemannian manifold. In this chapter we introduce the concept of geometric shape analysis, emphasizing the underlying geometric picture and how it relates to a variety of different ways to describe shapes. We describe the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework for shape deformation and show how the shape spaces inherit the geometry of the diffeomorphism group. For the common cases of landmarks and images, we describe the geodesic equations in explicit form. The setting allows us to perform statistical analysis on abstract representations of various types of shapes in one common geometric setting.

U2 - 10.1016/B978-0-12-814725-2.00011-X

DO - 10.1016/B978-0-12-814725-2.00011-X

M3 - Book chapter

SN - 978-0-12-814725-2

SP - 135

EP - 167

BT - Riemannian Geometric Statistics in Medical Image Analysis

A2 - Pennec, Xavier

A2 - Sommer, Stefan

A2 - Fletcher, Tom

PB - Academic Press

ER -