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)


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.


Convex optimisation Inverse processes Dynamical systems Diffusion filtering Swarm dynamics 



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.


  1. 1.
    Bergerhoff, L., Weickert, J.: Modelling image processing with discrete first-order swarms. In: Pillay, N., Engelbrecht, A.P., Abraham, A., du Plessis, M.C., Snášel, V., Muda, A.K. (eds.) Advances in Nature and Biologically Inspired Computing. AISC, vol. 419, pp. 261–270. Springer, Cham (2016). CrossRefGoogle Scholar
  2. 2.
    Carrillo, J.A., Fornasier, M., Toscani, G., Vecil, F.: Particle, kinetic, and hydrodynamic models of swarming. In: Naldi, G., Pareschi, L., Toscani, G. (eds.) Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, pp. 297–336. Birkhäuser, Boston (2010)CrossRefGoogle Scholar
  3. 3.
    Gilboa, G., Sochen, N.A., Zeevi, Y.Y.: Forward-and-backward diffusion processes for adaptive image enhancement and denoising. IEEE Trans. Image Process. 11(7), 689–703 (2002)CrossRefGoogle Scholar
  4. 4.
    Osher, S., Rudin, L.: Shocks and other nonlinear filtering applied to image processing. In: Tescher, A.G. (ed.) Proceedings of SPIE Applications of Digital Image Processing XIV, vol. 1567, pp. 414–431. SPIE Press, Bellingham (1991)Google Scholar
  5. 5.
    Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)CrossRefGoogle Scholar
  6. 6.
    Sapiro, G., Caselles, V.: Histogram modification via differential equations. J. Differ. Eqn. 135, 238–268 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Sochen, N.A., Zeevi, Y.Y.: Resolution enhancement of colored images by inverse diffusion processes. In: Proceeding of 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 2853–2856, Seattle, WA (May 1998)Google Scholar
  8. 8.
    Welk, M., Weickert, J.: An efficient and stable two-pixel scheme for 2D forward-and-backward diffusion. In: Lauze, F., Dong, Y., Dahl, A.B. (eds.) SSVM 2017. LNCS, vol. 10302, pp. 94–106. Springer, Cham (2017). CrossRefGoogle Scholar

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

Personalised recommendations