Martin Welk, Joachim Weickert, Guy Gilboa, “A Discrete Theory and Efficient Algorithms for Forward-and-Backward Diffusion Filtering”, preprint.
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 nonstandard 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
(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
Our experiments demonstrate that these algorithms allow speed-ups by up to three orders of magnitude without compromising stability or introducing visual artifacts.