Variational Methods in Shape Analysis

  • Martin Rumpf
  • Benedikt Wirth


The concept of a shape space is linked both to concepts from geometry and from physics. On one hand, a path-based viscous flow approach leads to Riemannian distances between shapes, where shapes are boundaries of objects that mainly behave like fluids. On the other hand, a state-based elasticity approach induces a (by construction) non-Riemannian dissimilarity measure between shapes, which is given by the stored elastic energy of deformations matching the corresponding objects. The two approaches are both based on variational principles. They are analyzed with regard to different applications, and a detailed comparison is given.


Hausdorff Distance Geodesic Distance Deformation Energy Dissimilarity Measure Shape Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The model proposed in Sect. 31.4.2 has been developed in cooperation with Leah Bar and Guillermo Sapiro from the University of Minnesota. Benedikt Wirth has been funded by the Bonn International Graduate School in Mathematics. Furthermore, the work was supported by the Deutsche Forschungsgemeinschaft, SPP 1253 “Optimization with Partial Differential Equations.” Part of Figs. 31-3 31-4 , and 31-19 31-23 have been taken from [83], the results from Figs. 31-6 , 31-8 , and 31-10 31-18 stem from [67, 69].

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Martin Rumpf
    • 1
  • Benedikt Wirth
    • 2
  1. 1.Bonn UniversityBonnGermany
  2. 2.Bonn UniversityBonnGermany

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