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Modelling Stable Backward Diffusion and Repulsive Swarms with Convex Energies and Range Constraints

  • Leif BergerhoffEmail author
  • Marcelo Cardénas
  • Joachim Weickert
  • Martin Welk
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10746)

Abstract

Backward diffusion and purely repulsive swarm dynamics are generally feared as ill-posed, highly unstable processes. On the other hand, it is well-known that minimising strictly convex energy functionals by gradient descent creates well-posed, stable evolutions. We prove a result that appears counterintuitive at first glance: We derive a class of one-dimensional backward evolutions from the minimisation of strictly convex energies. Moreover, we stabilise these inverse evolutions by imposing range constraints. This allows us to establish a comprehensive theory for the time-continuous evolution, and to prove a stability condition for an explicit time discretisation. Prototypical experiments confirm this stability and demonstrate that our model is useful for global contrast enhancement in digital greyscale images and for modelling purely repulsive swarm dynamics.

Keywords

Convex optimisation Inverse processes Dynamical systems Diffusion filtering Swarm dynamics 

Notes

Acknowledgement

Our research activities have been supported financially by the Deutsche Forschungsgemeinschaft (DFG) through a Gottfried Wilhelm Leibniz Prize for Joachim Weickert. This is gratefully acknowledged.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical Image Analysis GroupSaarland UniversitySaarbrückenGermany
  2. 2.Institute of Biomedical Image AnalysisPrivate University for Health Sciences, Medical Informatics and TechnologyHall/TyrolAustria

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