Related papers: On Regularization via Frame Decompositions with Ap…
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 characterize when a coherent state or continuous frame for a Hilbert space may be sampled to obtain a frame, which solves the discretization problem for continuous frames. In particular, we prove that every bounded continuous frame for a…
These lecture notes for a graduate class present the regularization theory for linear and nonlinear ill-posed operator equations in Hilbert spaces. Covered are the general framework of regularization methods and their analysis via spectral…
The reconstruction of images from their corresponding noisy Radon transform is a typical example of an ill-posed linear inverse problem as arising in the application of computerized tomography (CT). As the (naive) solution does not depend…
Despite the strong focus of regularization on ill-posed problems, the general construction of such methods has not been fully explored. Moreover, many previous studies cannot be clearly adapted to handle more complex scenarios, albeit the…
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…
Subspace recycling techniques have been used quite successfully for the acceleration of iterative methods for solving large-scale linear systems. These methods often work by augmenting a solution subspace generated iteratively by a known…
We consider an $\ell_1$-regularized inverse problem where both the forward and regularization operators have a Kronecker product structure. By leveraging this structure, a joint decomposition can be obtained using generalized singular value…
In this paper we propose a new class of iterative regularization methods for solving ill-posed linear operator equations. The prototype of these iterative regularization methods is in the form of second order evolution equation with a…
These lecture notes evolve around mathematical concepts arising in inverse problems. We start by introducing inverse problems through examples such as differentiation, deconvolution, computed tomography and phase retrieval. This then leads…
In this article, we study the convergence behavior of the regularization-based algorithm for solving the polynomial regression model when both input data and responses are from infinite-dimensional Hilbert spaces. We derive convergence…
We investigate the reconstruction problem of limited angle tomography. Such problems arise naturally in applications like digital breast tomosynthesis, dental tomography, electron microscopy etc. Since the acquired tomographic data is…
This manuscript is designed to introduce students in applied mathematics and data science to the concept of regularization for ill-posed inverse problems. Construct a mathematical model that describes how an image gets blurred. Convert a…
We present a family of non-local variational regularization methods for solving tomographic problems, where the solutions are functions with range in a closed subset of the Euclidean space, for example if the solution only attains values in…
We discuss the problem of estimating Radon-Nikodym derivatives. This problem appears in various applications, such as covariate shift adaptation, likelihood-ratio testing, mutual information estimation, and conditional probability…
We present a regularization method to approach a solution of the pessimistic formulation of ill -posed bilevel problems . This allows to overcome the difficulty arising from the non uniqueness of the lower level problems solutions and…
Let $X$ and $Y$ be Hilbert spaces, and $\mathbf{K}: \text{dom} \mathbf{K} \subset X \to Y$ a bounded linear operator. This paper addresses the inverse problem $\mathbf{K}x = y$, where exact data $y$ is replaced by noisy data $y^\delta$…
Image registration is fundamental in medical imaging applications, such as disease progression analysis or radiation therapy planning. The primary objective of image registration is to precisely capture the deformation between two or more…
In the context of linear inverse problems, we propose and study a general iterative regularization method allowing to consider large classes of regularizers and data-fit terms. The algorithm we propose is based on a primal-dual diagonal…
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…