Image enhancement wi th forward-and-backward (FAB) diffusion is numeric ally very challenging due to its negative diffusiv ities. As a remedy\, we first extend the explicit nonstandard scheme by Welk et al. (2009) from the 1D scenario to the practically relevant two-dimens ional setting. We prove that under a fairly severe time step restriction\, this 2D scheme preserves a maximum--minimum principle. Moreover\, we find a n interesting Lyapunov sequence which guarantees c onvergence to a flat steady state. Since a global application of the time step size restriction lead s to very slow algorithms and is more restrictive than necessary for most pixels\, we introduce a mu ch more efficient scheme with locally adapted time step sizes. It applies diffusive interactions of adjacent pixel pairs in a randomized order and ad apts the time step size locally. These space-varia nt time steps are synchronized at sync times which are determined by stability properties of the ex plicit forward diffusion scheme. Experiments show that our novel two-pixel scheme allows to compute FAB diffusion with guaranteed stability in the max imum norm at a speed that can be three orders of m agnitude larger than its explicit counterpart with a global time step size. \; LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR