Journal of Mathematical Imaging and Vision

, Volume 61, Issue 8, pp 1197–1220 | Cite as

Currents and Finite Elements as Tools for Shape Space

  • James Benn
  • Stephen MarslandEmail author
  • Robert I. McLachlan
  • Klas Modin
  • Olivier Verdier


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 as currents, which are based on linear spaces and are 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-based discretization of the currents that is a practical computational tool for shapes. Finally, we demonstrate this approach on a variety of examples.


Currents Finite elements Shape space Image analysis 

Mathematics Subject Classification

32U40m 62M40 65D18 74S05 



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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Palmerston NorthNew Zealand
  2. 2.School of Mathematics and StatisticsVictoria University of WellingtonWellingtonNew Zealand
  3. 3.Institute of Fundamental SciencesMassey UniversityPalmerston NorthNew Zealand
  4. 4.Department of Mathematical SciencesChalmers University of Technology and University of GothenburgGothenburgSweden
  5. 5.Department of Computing, Mathematics and PhysicsWestern Norway University of Applied SciencesBergenNorway
  6. 6.Department of MathematicsKTHStockholmSweden

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