We present and analyze a backtracking strategy for a general fast iterative shrinkage/thresholding algorithm proposed by Chambolle and Pock [Acta Numer., 25 (2016), pp. 161--319] for strongly conve...

Over the last 30 years a plethora of variational regularization models for image reconstruction have been proposed and thoroughly inspected by the applied mathematics community. Among them, the pioneering prototype often taught and learned in basic courses in mathematical image processing is the celebrated Rudin--Osher--Fatemi (ROF) model [L. I. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259--268], which relies on the minimization of the edge-preserving total variation (TV) seminorm as a regularization term. Despite its (often limiting) simplicity, this model is still very much employed in many applications and used as a benchmark for assessing the performance of modern learning-based image reconstruction approaches, thanks to its thorough analytical and numerical understanding. Among the many extensions to TV proposed over the years, a large class is based on the concept of space variance. Space-variant models can indeed overcome the intrinsic inability of TV to describe local features (strength, sharpness, directionality) by means of an adaptive mathematical modeling which accommodates local regularization weighting, variable smoothness, and anisotropy. Those ideas can further be cast in the flexible Bayesian framework of generalized Gaussian distributions and combined with maximum likelihood and hierarchical optimization approaches for efficient hyperparameter estimation. In this work, we review and connect the major contributions in the field of space-variant TV-type image reconstruction models, focusing, in particular, on their Bayesian interpretation which paves the way to new exciting and unexplored research directions.

We consider the problem of scale detection in images where a region of interest is present together with a measurement tool (e.g. a ruler). For the segmentation part, we focus on the graph-based method presented in Bertozzi and Flenner (Multiscale Model Simul 10(3):1090–1118, 2012) which reinterprets classical continuous Ginzburg–Landau minimisation models in a totally discrete framework. To overcome the numerical difficulties due to the large size of the images considered, we use matrix completion and splitting techniques. The scale on the measurement tool is detected via a Hough transform-based algorithm. The method is then applied to some measurement tasks arising in real-world applications such as zoology, medicine and archaeology.

We propose a new variational model for non-linear image fusion. Our approach is based on the use of an osmosis energy term related to the one studied in Vogel et al. and Weickert et al. The minimization of the proposed non-convex energy realizes visually plausible image data fusion, invariant to multiplicative brightness changes. On the practical side, it requires minimal supervision and parameter tuning and can encode prior information on the structure of the images to be fused. For the numerical solution of the proposed model, we develop a primal-dual algorithm and we apply the resulting minimization scheme to solve multi-modal face fusion, color transfer and cultural heritage conservation problems. Visual and quantitative comparisons to state-of-the-art approaches prove the out-performance and the flexibility of our method.

We present an anisotropic extension of the isotropic osmosis model that has been introduced by Weickert et al (2013 Energy Minimization Methods in Computer Vision and Pattern Recognition (Berlin: Springer)) for visual computing applications, and we adapt it specifically to shadow removal applications. We show that in the integrable setting, linear anisotropic osmosis minimises an energy that involves a suitable quadratic form which models local directional structures. In our shadow removal applications we estimate the local structure via a modified tensor voting approach (Moreno et al 2012 New Developments in the Visualization and Processing of Tensor Fields (Berlin: Springer)) and use this information within an anisotropic diffusion inpainting that resembles edge-enhancing anisotropic diffusion inpainting (Galić et al 2008 J. Math. Imaging Vis . 31 255–69; Weickert and Welk 2006 Visualization and Processing of Tensor Fields (Berlin: Springer)). Our numerical scheme combines the nonnegativity preserving stencil of Fehrenbach and Mirebeau (2014 J. Math. Imaging Vis . 49 123–47) with an exact time stepping based on highly accurate polynomial approximations of the matrix exponential. The resulting anisotropic model is tested on several synthetic and natural images corrupted by constant shadows. We show that it outperforms isotropic osmosis, since it does not suffer from blurring artefacts at the shadow boundaries.

We show that the Wilson–Cowan equations can reproduce a number of brightness and orientation-dependent illusions. Then we formally prove that there cannot be an energy functional that the Wilson–Cowan equations are minimizing, making them suboptimal with respect to the efficient representation principle. We thus propose a slight modification that is consistent with such principle and show that this provides a better reproduction of visual illusions than the original Wilson–Cowan formulation. We also consider the cortical extension of both models to deal with more complex orientation-dependent illusions.