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

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VMVW01 - Variational methods, new optimisation techniques and new fast numerical algorithms

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. 

This talk is part of the Isaac Newton Institute Seminar Series series.

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