# Currents and finite elements as tools for shape space

## TL;DR

This paper introduces a new approach for representing and analyzing shapes using currents and finite element methods, enabling robust numerical discretization and shape comparison in nonlinear shape spaces.

## Contribution

It develops the theory of currents for shape spaces, analyzes their properties, and provides a finite element discretization for practical shape analysis.

## Key findings

- Current map determines shapes under certain conditions.
- Finite element discretization is effective for shape computation.
- Approach is robust to noise and suitable for numerical implementation.

## Abstract

The nonlinear spaces of shapes (unparameterized immersed curves or submanifolds) are of interest for many applications in image analysis, such as the 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 on linear spaces and is suitable for numerical discretization, being robust to noise. We develop the theory of currents for shape spaces by considering both the analytic and numerical aspects of the problem. In particular, we study the analytical properties of the current map and the $H^{-s}$ norm that it induces on shapes. We determine the conditions under which the current determines the shape. We then provide a finite element discretization of the currents that is a practical computational tool for shapes. Finally, we demonstrate this approach on a variety of examples.