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.

Tags: discrete scheme , FAB , nonlinear diffusion

Image above: Eigenfunction of TGV found by the proposed flow.

Abstract:

Nonlinear variational methods have become very powerful tools for many image processing tasks. Recently a new line of research has emerged, dealing with nonlinear eigenfunctions induced by convex functionals. This has provided new insights and better theoretical understanding of convex regularization and introduced new processing methods. However, the theory of nonlinear eigenvalue problems is still at its infancy. We present a new flow that can generate nonlinear eigenfunctions of the form $T(u)=\lambda u$, where $T(u)$ is a nonlinear operator and $\lambda \in \mathbb{R} $ is the eigenvalue. We develop the theory where $T(u)$ is a subgradient element of a regularizing one-homogeneous functional, such as total-variation (TV) or total-generalized-variation (TGV). We introduce two flows: a forward flow and an inverse flow; for which the steady state solution is a nonlinear eigenfunction. The forward flow monotonically smooths the solution (with respect to the regularizer) and simultaneously increases the $L^2$ norm. The inverse flow has the opposite characteristics. For both flows, the steady state depends on the initial condition, thus different initial conditions yield different eigenfunctions. This enables a deeper investigation into the space of nonlinear eigenfunctions, allowing to produce numerically diverse examples, which may be unknown yet. In addition we suggest an indicator to measure the affinity of a function to an eigenfunction and relate it to pseudo-eigenfunctions in the linear case

Ester Hait, Guy Gilboa, “Blind Facial Image Quality Enhancement using Non-Rigid Semantic Patches”, accepted to IEEE Trans. Image Processing, 2017.

Abstract:

We propose to combine semantic data and registration algorithms to solve various image processing problems such as denoising, super-resolution and color-correction.
It is shown how such new techniques can achieve significant quality enhancement, both visually and quantitatively, in the case of facial image enhancement. Our model assumes prior high quality data of the person to be processed, but no knowledge of the degradation model.
We try to overcome the classical processing limits by using semantically-aware patches, with adaptive size and location regions of coherent structure and context, as building blocks. The method is demonstrated on the problem of cellular photography enhancement of dark facial images for different identities, expressions and poses