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Inverse problems are fundamental in fields like medical imaging, geophysics, and computerized tomography, aiming to recover unknown quantities from observed data. However, these problems often lack stability due to noise and…

Inverse problems arise in a number of domains such as medical imaging, remote sensing, and many more, relying on the use of advanced signal and image processing approaches -- such as sparsity-driven techniques -- to determine their…

Inverse problems are inherently ill-posed and therefore require regularization techniques to achieve a stable solution. While traditional variational methods have well-established theoretical foundations, recent advances in machine learning…

The problem of object restoration in the case of spatially incoherent illumination is considered. A regularized solution to the inverse problem is obtained through a probabilistic approach, and a numerical algorithm based on the statistical…

When solving rank-deficient or discrete ill-posed problems by regularization methods, the choice of the regularization parameter is crucial. It is also of interest, the regularization norm used in the selection of the solution. In this…

In this work, we investigate data fitting problems with random noises. A randomized progressive iterative regularization method is proposed. It works well for large-scale matrix computations and converges in expectation to the least-squares…

Inverse problems lie at the heart of modern imaging science, with broad applications in areas such as medical imaging, remote sensing, and microscopy. Recent years have witnessed a paradigm shift in solving imaging inverse problems, where…

It is of importance to develop statistical techniques to analyze high-dimensional data in the presence of both complex dependence and possible outliers in real-world applications such as imaging data analyses. We propose a new robust…

Data-driven approaches recently achieved remarkable success in magnetic resonance imaging (MRI) reconstruction, but integration into clinical routine remains challenging due to a lack of generalizability and interpretability. In this paper,…

We investigate continuous regularization methods for linear inverse problems of static and dynamic type. These methods are based on dynamic programming approaches for linear quadratic optimal control problems. We prove regularization…

Solving inverse problems requires appropriate regularization techniques to ensure well-posedness and stability. In recent years, denoiser-driven methods have emerged as effective regularization strategies, achieving state-of-the-art…

In this article we investigate the connection between regularization theory for inverse problems and dynamic programming theory. This is done by developing two new regularization methods, based on dynamic programming techniques. The aim of…

We propose a novel data-dependent structured gradient regularizer to increase the robustness of neural networks vis-a-vis adversarial perturbations. Our regularizer can be derived as a controlled approximation from first principles,…

One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization…

Low-rank structures play important role in recent advances of many problems in image science and data science. As a natural extension of low-rank structures for data with nonlinear structures, the concept of the low-dimensional manifold…

When adopting a model-based formulation, solving inverse problems encountered in multiband imaging requires to define spatial and spectral regularizations. In most of the works of the literature, spectral information is extracted from the…

We propose an adaptive regularization scheme in a variational framework where a convex composite energy functional is optimized. We consider a number of imaging problems including denoising, segmentation and motion estimation, which are…

Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, deconvolution, inpainting, compressed sensing, and superresolution all lie in this framework. Traditional inverse problem solvers…

Learning-based methods for inverse problems, adapting to the data's inherent structure, have become ubiquitous in the last decade. Besides empirical investigations of their often remarkable performance, an increasing number of works…

We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear…