A Discrete Theory and Efficient Algorithms for Forward-and-Backward Diffusion Filtering

Image enhancement with forward-and-backward (FAB) diffusion lacks a sound theory and is numerically very challenging due to its diffusivities that are negative within a certain gradient range. In our paper, we address both problems. First we establish a comprehensive theory for space-discrete and time-continuous FAB diffusion processes. It requires approximating the gradient magnitude with a non-standard discretisation. Then, we show that this theory carries over to the fully discrete case, when an explicit time discretisation with a fairly restrictive step-size limit is applied. To come up with more efficient algorithms, we propose three accelerated schemes: (i) an explicit scheme with global time step size adaptation that is also well suited for parallel implementations on GPUs, (ii) a randomised two-pixel scheme that offers optimal adaptivity of the time step size, (iii) a deterministic two-pixel scheme which benefits from less restrictive consistency bounds. Our experiments demonstrate that these algorithms allow speed-ups by up to three orders of magnitude without compromising stability or introducing visual artefacts.