skip to content

Efficient and Stable Schemes for 2D Forward-and-Backward Diffusion

Presented by: 
Joachim Weickert
Monday 4th September 2017 - 09:50 to 10:40
INI Seminar Room 1
Co-author: Martin Welk (UMIT Hall, Austria)

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 interactions of adjacent pixel pairs in a randomized order and adapts the time step size locally. These space-variant time steps are synchronized at sync times which are determined by stability properties of the explicit forward diffusion scheme. Experiments show that our novel two-pixel scheme allows to compute FAB diffusion with guaranteed stability in the maximum norm at a speed that can be three orders of magnitude larger than its explicit counterpart with a global time step size. 
The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons