Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known...

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Bibliographic Details
Published inJournal of mathematical imaging and vision Vol. 50; no. 1-2; pp. 60 - 97
Main Authors Bauer, Martin, Bruveris, Martins, Michor, Peter W.
Format Journal Article
LanguageEnglish
Published Boston Springer US 01.09.2014
Subjects
Applications of Mathematics
Computer Science
Image Processing and Computer Vision
Mathematical Methods in Physics
Signal,Image and Speech Processing
Riemannian geometry
Shape space
Manifolds of mappings
Surface matching
Diffeomorphism group
Landmark space
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Summary:This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature.
ISSN:0924-9907
1573-7683
DOI:10.1007/s10851-013-0490-z