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This is because curves are formally represented as functions, and spaces of functions are usually infinite © Springer-Verlag New York 2016 A. P. 1007/978-1-4939-4020-2 3 39 40 3 Background: Relevant Tools from Geometry dimensional. To analyze these spaces of functions, we require some tools from functional analysis. Although computer experiments will ultimately involve discretizing these functions into finite sets of points, the basic theory shall be developed assuming continuous, or functional, representations.
22 Symmetry analysis: What is the amount of asymmetry in these shapes? Fig. 23 A system of statistical analysis of shapes in images We will develop a formal approach for analyzing symmetry of objects using deformations; here the objects are deformed until they become symmetric and the amount of deformation measures the level of asymmetry. In summary, our goal is to develop a fully statistical framework in which we can treat shapes of functions and curves as random variables taking values in welldefined shape spaces and governed by underlying probability densities.
This is because curves are formally represented by functions, and spaces of functions are usually infinite dimensional. Although computer experiments will ultimately involve discretization of these functions into finite sets of points, the basic theory shall be developed assuming continuous, or functional, representations. 5 Organization of this Textbook 19 In summary, the nonlinearity and the infinite dimensionality of shapes, and the need for certain invariances, make it difficult to perform shape analysis of curves.
