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We propose a new regularisation strategy for the classical ensemble Kalman inversion (EKI) framework. The strategy consists of: (i) an adaptive choice for the regularisation parameter in the update formula in EKI, and (ii) criteria for the…
The new version of a posteriori choice (NVAC) of the regularization parameter in the classical Tikhonov regularization method is considered. Lemmas and theorems on the error and the asymptotic convergence rate of the regularized solution…
The reconstruction of low-rank matrix from its noisy observation finds its usage in many applications. It can be reformulated into a constrained nuclear norm minimization problem, where the bound $\eta$ of the constraint is explicitly given…
Tomographic reconstruction is an ill-posed inverse problem that calls for regularization. One possibility is to require sparsity of the unknown in an orthonormal wavelet basis. This in turn can be achieved by variational regularization…
It is well-known that point sources with sufficient mutual distance can be reconstructed exactly from finitely many Fourier measurements by solving a convex optimization problem with Tikhonov-regularization (this property is sometimes…
Many inverse problems include nuisance parameters which, while not of direct interest, are required to recover primary parameters. Structure present in these problems allows efficient optimization strategies - a well known example is…
In this work we derive higher order error estimates for inverse problems distorted by non-additive noise, in terms of Bregman distances. The results are obtained by means of a novel source condition, inspired by the dual problem.…
Compressed sensing (CS) methods in magnetic resonance imaging (MRI) offer rapid acquisition and improved image quality but require iterative reconstruction schemes with regularization to enforce sparsity. Regardless of the difficulty in…
This paper considers large-scale linear ill-posed inverse problems whose solutions can be represented as sums of smooth and piecewise constant components. To solve such problems we consider regularizers consisting of two terms that must be…
In many applications such as medical imaging, the measurement data represent counts of photons hitting a detector. Such counts in low-photon settings are often modeled using a Poisson distribution. However, this model assumes that the mean…
This study presents the development of a spatially adaptive weighting strategy for Total Variation regularization, aimed at addressing under-determined linear inverse problems. The method leverages the rapid computation of an accurate…
A novel class of semi-norms, generalising the notion of the isotropic total variation $TV_{2}$ and the an-isotropic total variation $TV_{1}$ is introduced. A supervised learning method via bilevel optimisation is proposed for the…
In this paper, we propose a Bayesian MAP estimator for solving the deconvolution problems when the observations are corrupted by Poisson noise. Towards this goal, a proper data fidelity term (log-likelihood) is introduced to reflect the…
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…
We present a new vectorial total variation method that addresses the problem of color consistent image filtering. Our approach is inspired from the double-opponent cell representation in the human visual cortex. Existing methods of…
Tikhonov regularization is one of the most commonly used methods of regularization of ill-posed problems. In the setting of finite element solutions of elliptic partial differential control problems, Tikhonov regularization amounts to…
We introduce a unified framework based on bi-level optimization schemes to deal with parameter learning in the context of image processing. The goal is to identify the optimal regularizer within a family depending on a parameter in a…
Total variation (TV) regularization is a popular reconstruction method for ill-posed imaging problems, and particularly useful for applications with piecewise constant targets. However, using TV for medical cone-beam computed X-ray…
Deep neural networks achieve high prediction accuracy when the train and test distributions coincide. In practice though, various types of corruptions occur which deviate from this setup and cause severe performance degradations. Few…
We present a fast algorithm for the total variation regularization of the $3$-D gravity inverse problem. Through imposition of the total variation regularization, subsurface structures presenting with sharp discontinuities are preserved…