The Measurement and Analysis of Shapes

Abstract

A de Rham p-current can be viewed as a map (the current map) between the set of embeddings of a closed p-dimensional manifold into an ambient n-manifold and the set of linear functionals on differential p-forms. We demonstrate that, for suitably chosen Sobolev topologies on both the space of embeddings and the space of p-forms, the current map is continuously differentiable, with an image that consists of bounded linear functionals on p-forms. Using the Riesz representation theorem, we prove that each p-current can be represented by a unique co-exact differential form that has a particular interpretation depending on p. Embeddings of a manifold can be thought of as shapes with a prescribed topology. Our analysis of the current map provides us with representations of shapes that can be used for the measurement and statistical analysis of collections of shapes. We consider two special cases of our general analysis and prove that: (1) if p=n-1 then closed, embedded, co-dimension one surfaces are naturally represented by probability distributions on the ambient manifold and (2) if p=1 then closed, embedded, one-dimensional curves are naturally represented by fluid flows on the ambient manifold. In each case, we outline some statistical applications using an {dot{H}}^{1} and L^{2} metric, respectively.