An Efficient and Stable Two-Pixel Scheme for 2D Forward-and-Backward Diffusion

  • Martin WelkEmail author
  • Joachim Weickert
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10302)


Image enhancement with forward-and-backward (FAB) diffusion is numerically very challenging due to its negative diffusivities. As a remedy, we first extend the explicit nonstandard scheme by Welk et al. (2009) from the 1D scenario to the practically relevant two-dimensional setting. We prove that under a fairly severe time step restriction, this 2D scheme preserves a maximum–minimum principle. Moreover, we find an interesting Lyapunov sequence which guarantees convergence to a flat steady state. Since a global application of the time step size restriction leads to very slow algorithms and is more restrictive than necessary for most pixels, we introduce a much more efficient scheme with locally adapted time step sizes. It applies diffusive two-pixel interactions in a randomised order and adapts the time step size to the specific pixel pair. These space-variant time steps are synchronised at sync times. Our experiments show that our novel two-pixel scheme allows to compute FAB diffusion with guaranteed \(L^\infty \)-stability at a speed that can be three orders of magnitude larger than its explicit counterpart with a global time step size.


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© Springer International Publishing AG 2017

Authors and Affiliations

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

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