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We propose an unsupervised approach for learning end-to-end reconstruction operators for ill-posed inverse problems. The proposed method combines the classical variational framework with iterative unrolling, which essentially seeks to…

Computer Vision and Pattern Recognition · Computer Science 2021-06-08 Subhadip Mukherjee , Marcello Carioni , Ozan Öktem , Carola-Bibiane Schönlieb

This paper discusses basic results and recent developments on variational regularization methods, as developed for inverse problems. In a typical setup we review basic properties needed to obtain a convergent regularization scheme and…

Machine Learning · Computer Science 2021-12-10 Martin Burger

In this work, we propose a framework to learn a local regularization model for solving general image restoration problems. This regularizer is defined with a fully convolutional neural network that sees the image through a receptive field…

Image and Video Processing · Electrical Eng. & Systems 2021-02-12 Jean Prost , Antoine Houdard , Andrés Almansa , Nicolas Papadakis

In this work, we consider the linear inverse problem $y=Ax+\epsilon$, where $A\colon X\to Y$ is a known linear operator between the separable Hilbert spaces $X$ and $Y$, $x$ is a random variable in $X$ and $\epsilon$ is a zero-mean random…

Machine Learning · Statistics 2021-11-23 Giovanni S. Alberti , Ernesto De Vito , Matti Lassas , Luca Ratti , Matteo Santacesaria

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…

Optimization and Control · Mathematics 2021-01-27 S. Kindermann , A. Leitao

This review provides an introduction to - and overview of - the current state of the art in neural-network based regularization methods for inverse problems in imaging. It aims to introduce readers with a solid knowledge in applied…

Optimization and Control · Mathematics 2023-12-25 Andreas Habring , Martin Holler

Variational regularization has remained one of the most successful approaches for reconstruction in imaging inverse problems for several decades. With the emergence and astonishing success of deep learning in recent years, a considerable…

Machine Learning · Computer Science 2021-10-26 Subhadip Mukherjee , Carola-Bibiane Schönlieb , Martin Burger

We provide estimators for a large class of inverse problems, including nonlinear inverse problems. Using complexity regularization technics we provide adaptive estimators achieving the best rate over the collection of models.

Statistics Theory · Mathematics 2007-06-13 Jean Michel Loubes , Ludeña Carenne

Objective: This paper investigates how generative models, trained on ground-truth images, can be used \changes{as} priors for inverse problems, penalizing reconstructions far from images the generator can produce. The aim is that learned…

Image and Video Processing · Electrical Eng. & Systems 2023-08-07 Margaret Duff , Ivor J. A. Simpson , Matthias J. Ehrhardt , Neill D. F. Campbell

This article addresses the challenge of learning effective regularizers for linear inverse problems. We analyze and compare several types of learned variational regularization against the theoretical benchmark of the optimal affine…

Numerical Analysis · Mathematics 2025-10-15 Sebastian Banert , Christoph Brauer , Dirk Lorenz , Lionel Tondji

Discrete inverse problems correspond to solving a system of equations in a stable way with respect to noise in the data. A typical approach to enforce uniqueness and select a meaningful solution is to introduce a regularizer. While for most…

Optimization and Control · Mathematics 2022-04-22 Cristian Vega , Cesare Molinari , Lorenzo Rosasco , Silvia Villa

Neural networks have emerged as effective tools for solving ill-posed inverse problems. In many scientific applications, however, observational training data are insufficient, and learned inverse operators must instead be trained on…

Numerical Analysis · Mathematics 2026-05-26 Sandra R. Babyale , Jodi Mead

Recently, with the significant developments in deep learning techniques, solving underdetermined inverse problems has become one of the major concerns in the medical imaging domain. Typical examples include undersampled magnetic resonance…

Image and Video Processing · Electrical Eng. & Systems 2020-06-29 Chang Min Hyun , Seong Hyeon Baek , Mingyu Lee , Sung Min Lee , Jin Keun Seo

Learning neural networks using only few available information is an important ongoing research topic with tremendous potential for applications. In this paper, we introduce a powerful regularizer for the variational modeling of inverse…

Machine Learning · Computer Science 2023-05-17 Fabian Altekrüger , Alexander Denker , Paul Hagemann , Johannes Hertrich , Peter Maass , Gabriele Steidl

Theoretical guarantees for the robust solution of inverse problems have important implications for applications. To achieve both guarantees and high reconstruction quality, we propose learning a pixel-based ridge regularizer with a…

Optimization and Control · Mathematics 2025-01-07 Sebastian Neumayer , Fabian Altekrüger

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…

Numerical Analysis · Mathematics 2022-05-27 Alessandro Benfenati , Giuseppe Bisazza , Paola Causin

Learned inverse problem solvers exhibit remarkable performance in applications like image reconstruction tasks. These data-driven reconstruction methods often follow a two-step scheme. First, one trains the often neural network-based…

Recently, data-driven techniques have demonstrated remarkable effectiveness in addressing challenges related to MR imaging inverse problems. However, these methods still exhibit certain limitations in terms of interpretability and…

Computer Vision and Pattern Recognition · Computer Science 2023-09-19 Huayu Wang , Chen Luo , Taofeng Xie , Qiyu Jin , Guoqing Chen , Zhuo-Xu Cui , Dong Liang

Variational regularization of ill-posed inverse problems is based on minimizing the sum of a data fidelity term and a regularization term. The balance between them is tuned using a positive regularization parameter, whose automatic choice…

Numerical Analysis · Mathematics 2025-11-12 Markus Juvonen , Bjørn Jensen , Ilmari Pohjola , Yiqiu Dong , Samuli Siltanen

How to extract more and useful information for single image super resolution is an imperative and difficult problem. Learning-based method is a representative method for such task. However, the results are not so stable as there may exist…

Image and Video Processing · Electrical Eng. & Systems 2020-03-25 Hu Liang , Shengrong Zhao