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This paper is concerned with a novel regularisation technique for solving linear ill-posed operator equations in Hilbert spaces from data that is corrupted by white noise. We combine convex penalty functionals with extreme-value statistics…
This work considers using reduced basis techniques in connection to (smoothened) total variation regularization in electrical impedance tomography, but analogous ideas can also be used for other inverse elliptic boundary value problems. It…
Robustness and stability of image-reconstruction algorithms have recently come under scrutiny. Their importance to medical imaging cannot be overstated. We review the known results for the topical variational regularization strategies…
We consider inverse problems with large null spaces, which arise in important applications such as in inverse ECG and EEG procedures. Standard regularization methods typically produce solutions in or near the orthogonal complement of the…
Conditional stability estimates require additional regularization for obtaining stable approximate solutions if the validity area of such estimates is not completely known. In this context, we consider ill-posed nonlinear inverse problems…
The estimation of motion in an image sequence is a fundamental task in image processing. Frequently, the image sequence is corrupted by noise and one simultaneously asks for the underlying motion field and a restored sequence. In smoothly…
It's well-known that inverse problems are ill-posed and to solve them meaningfully one has to employ regularization methods. Traditionally, popular regularization methods have been the penalized Variational approaches. In recent years, the…
We address the image restoration problem under Poisson noise corruption. The Kullback-Leibler divergence, which is typically adopted in the variational framework as data fidelity term in this case, is coupled with the second-order Total…
We propose a second-order total generalized variation (TGV) regularization for the reconstruction of the initial condition in variational data assimilation problems. After showing the equivalence between TGV regularization and the Bayesian…
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…
In this paper, we propose a new framework to remove parts of the systematic errors affecting popular restoration algorithms, with a special focus for image processing tasks. Generalizing ideas that emerged for $\ell_1$ regularization, we…
In this paper we study the structure of solutions of the one dimensional weighted total variation regularisation problem, motivated by its application in signal recovery tasks. We study in depth the relationship between the weight function…
We study the solutions of infinite dimensional linear inverse problems over Banach spaces. The regularizer is defined as the total variation of a linear mapping of the function to recover, while the data fitting term is a near arbitrary…
Inverse Problems in medical imaging and computer vision are traditionally solved using purely model-based methods. Among those variational regularization models are one of the most popular approaches. We propose a new framework for applying…
Solving inverse problems \(Ax = y\) is central to a variety of practically important fields such as medical imaging, remote sensing, and non-destructive testing. The most successful and theoretically best-understood method is convex…
Many imaging problems require solving an inverse problem that is ill-conditioned or ill-posed. Imaging methods typically address this difficulty by regularising the estimation problem to make it well-posed. This often requires setting the…
The total variation-based image denoising model has been generalized and extended in numerous ways, improving its performance in different contexts. We propose a new penalty function motivated by the recent progress in the statistical…
This paper focuses on the development of a space-variant regularization model for solving an under-determined linear inverse problem. The case study is a medical image reconstruction from few-view tomographic noisy data. The primary…
In this paper, we establish robustness to noise perturbations of polyhedral regularization of linear inverse problems. We provide a sufficient condition that ensures that the polyhedral face associated to the true vector is equal to that of…
We propose a new type of regularization functional for images called oscillation total generalized variation (TGV) which can represent structured textures with oscillatory character in a specified direction and scale. The infimal…