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

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
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 artifacts.

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