Abstract
Shape spaces are typically nonlinear spaces where the rules of vector-space calculus do not apply directly. Instead, one uses tools from algebra, geometry, and functional analysis to develop frameworks for optimization and statistics in shape spaces. In this chapter, we provide a brief introduction of fundamental tools and techniques from algebra, differential geometry and functional analysis that are basic to shape analysis. This topic includes definitions and examples of groups, differentiable manifolds and actions of groups on manifolds. For understanding differentiable geometry of shape manifolds, we introduce the concepts of tangent spaces, Riemannian structures, geodesics, and exponential maps. An important development in shape analysis of curves has been the involvement of functional analysis. The representations of continuous curves involve functions on real lines or planes, and we will summarize basic concepts from functional analysis. In particular, we will establish the notions of Hilbert spaces, submanifolds, …
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Srivastava, A., Klassen, E.P. (2016). Background: Relevant Tools from Geometry. In: Functional and Shape Data Analysis. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-4020-2_3
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DOI: https://doi.org/10.1007/978-1-4939-4020-2_3
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