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The nonlinear spaces of shapes (unparameterized immersed curves orsubmanifolds) are of interest for many applications in image analysis, such asthe identification of shapes that are similar modulo the action of some group.In this paper we study a general representation of shapes that is based onlinear spaces and is suitable for numerical discretization, being robust tonoise. We develop the theory of currents for shape spaces by considering boththe analytic and numerical aspects of the problem. In particular, we study theanalytical properties of the current map and the $H^{-s}$ norm that it induceson shapes. We determine the conditions under which the current determines theshape. We then provide a finite element discretization of the currents that isa practical computational tool for shapes. Finally, we demonstrate thisapproach on a variety of examples.

Currents and finite elements as tools for shape space