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Diffuse Optical Tomography (DOT) is an emerging technology in medical imaging which employs light in the NIR spectrum to estimate the distribution of optical coefficients in biological tissues for diagnostic and monitoring purposes. DOT…

Over the last decades, the total variation (TV) evolved to one of the most broadly-used regularisation functionals for inverse problems, in particular for imaging applications. When first introduced as a regulariser, higher-order…

Differential flatness serves as a powerful tool for controlling continuous time nonlinear systems in problems such as motion planning and trajectory tracking. A similar notion, called difference flatness, exists for discrete-time systems.…

In an attempt to better understand generalization in deep learning, we study several possible explanations. We show that implicit regularization induced by the optimization method is playing a key role in generalization and success of deep…

The main goal of this paper is to provide a brief survey of recent results which connect together results from different areas of research. It is well known that numerical integration of functions with mixed smoothness is closely related to…

Based on a nonsmooth coherence condition, we construct and prove the convergence of a forward-backward splitting method that alternates between steps on a fine and a coarse grid. Our focus is a total variation regularised inverse imaging…

In this paper, a new regularization term is proposed to solve mathematical image problems. By using difference operators in the four directions; horizontal, vertical and two diagonal directions, an estimation of derivative amplitude is…

The success of denoising diffusion models raises important questions regarding their generalisation behaviour, particularly in high-dimensional settings. Notably, it has been shown that when training and sampling are performed perfectly,…

Variational regularization models are one of the popular and efficient approaches for image restoration. The regularization functional in the model carries prior knowledge about the image to be restored. The prior knowledge, in particular…

We study the generalization of over-parameterized deep networks (for image classification) in relation to the convex hull of their training sets. Despite their great success, generalization of deep networks is considered a mystery. These…

Machine learning is predicated on the concept of generalization: a model achieving low error on a sufficiently large training set should also perform well on novel samples from the same distribution. We show that both data whitening and…

In a series of papers (Lombardi & Schneider 2001, 2002) we studied in detail the statistical properties of an interpolation technique widely used in astronomy. In particular, we considered the average interpolated map and its covariance…

In this paper, a new general decomposition theory inspired from modular graph decomposition is presented. Our main result shows that, within this general theory, most of the nice algorithmic tools developed for modular decomposition are…

The total variation (TV) regularization has phenomenally boosted various variational models for image processing tasks. We propose to combine the backward diffusion process in the earlier literature of image enhancement with the TV…

A Bayesian hierarchical model for total variation regularisation is presented in this paper. All the parameters of an inverse problem, including the "regularisation parameter", are estimated simultaneously from the data in the model. The…

In this paper we study a variational problem in the space of functions of bounded Hessian. Our model constitutes a straightforward higher-order extension of the well known ROF functional (total variation minimisation) to which we add a…

In this paper we investigate an adaptive discretization strategy for ill-posed linear prob- lems combined with a regularization from a class of semiiterative methods. We show that such a discretization approach in combination with a…

In this paper, we propose a second-order dynamical system with a smoothing effect for solving paramonotone variational inequalities. Under standard assumptions, we prove that the trajectories of this dynamical system converges to a solution…

The characteristic feature of inverse problems is their instability with respect to data perturbations. In order to stabilize the inversion process, regularization methods have to be developed and applied. In this work we introduce and…

We propose an extension of the discretization approaches for multilayer shallow water models, aimed at making them more flexible and efficient for realistic applications to coastal flows. A novel discretization approach is proposed, in…