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∞ -stability at a speed that can be three orders of magnitude larger than its explicit counterpart with a global time step size.
CITATION STYLE
Welk, M., & Weickert, J. (2017). An efficient and stable two-pixel scheme for 2D forward-and-backward diffusion. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 10302 LNCS, pp. 94–106). Springer Verlag. https://doi.org/10.1007/978-3-319-58771-4_8
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