Forward-and-backward (FAB) diffusion is a method for sharpening blurry images (Gilboa et al. 2002). It combines forward diffusion with a positive diffusivity and backward diffusion where negative diffusivities are used. The well-posedness properties of FAB diffusion are unknown, and it has been observed that standard discretisations can violate a maximum-minimum principle. We show that for a novel nonstandard space discretisation which pays specific attention to image extrema, one can apply a modification of the space-discrete well-posedness and scale-space framework of Weickert (1998). This allows to establish well-posedness and a maximum-minimum principle for the resulting dynamical system. In the fully discrete 1-D case with an explicit time discretisation, a maximum-minimum principle and total variation reduction are proven in spite of the fact that negative diffusivities may appear. This provides a theoretical justification for applying FAB diffusion to digital images. © 2009 Springer Berlin Heidelberg.
CITATION STYLE
Welk, M., Gilboa, G., & Weickert, J. (2009). Theoretical foundations for discrete forward-and-backward diffusion filtering. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 5567 LNCS, pp. 527–538). https://doi.org/10.1007/978-3-642-02256-2_44
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