Variational regularisation for inverse problems with imperfect forward operators and general noise models

Bungert, Leon; Burger, Martin; Korolev, Yury; Schönlieb, Carola‐Bibiane

doi:10.1088/1361-6420/abc531

Cited by 12 publications

(5 citation statements)

References 52 publications

(92 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 4.20. The same rates can be obtained for the symmetric Bregman distance (see, e.g., [47,Thm. 3.6]).…”

Section: Convergence Rates In Bregman Distancesupporting

confidence: 63%

Two-layer neural networks with values in a Banach space

Korolev¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We study two-layer neural networks whose domain and range are Banach spaces with separable preduals. In addition, we assume that the image space is equipped with a partial order, i.e. it is a Riesz space. As the nonlinearity we choose the lattice operation of taking the positive part; in case of R d -valued neural networks this corresponds to the ReLU activation function. We prove inverse and direct approximation theorems with Monte-Carlo rates, extending existing results for the finite-dimensional case. In the second part of the paper, we consider training such networks using a finite amount of noisy observations from the regularisation theory viewpoint. We discuss regularity conditions known as source conditions and obtain convergence rates in a Bregman distance in the regime when both the noise level goes to zero and the number of samples goes to infinity at appropriate rates.

show abstract

“…Remark 4.20. The same rates can be obtained for the symmetric Bregman distance (see, e.g., [47,Thm. 3.6]).…”

Section: Convergence Rates In Bregman Distancesupporting

confidence: 63%

Two-layer neural networks with values in a Banach space

Korolev¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Recent studies have investigated the effect of using approximate or imperfect operators in the reconstruction [49]- [51] and the possibility to improve results with a learned correction [15], [16]. Most importantly, it is shown that under a suitable learned correction, one can accurately solve the classic variational problem.…”

Section: B a Fast Approximate Forward And Inverse Modelmentioning

confidence: 99%

Model-Based Learning for Accelerated, Limited-View 3-D Photoacoustic Tomography

Hauptmann

Lucka

Betcke

et al. 2018

IEEE Trans. Med. Imaging

293

246

View full text Add to dashboard Cite

Recent advances in deep learning for tomographic reconstructions have shown great potential to create accurate and high quality images with a considerable speed up. In this paper, we present a deep neural network that is specifically designed to provide high resolution 3-D images from restricted photoacoustic measurements. The network is designed to represent an iterative scheme and incorporates gradient information of the data fit to compensate for limited view artifacts. Due to the high complexity of the photoacoustic forward operator, we separate training and computation of the gradient information. A suitable prior for the desired image structures is learned as part of the training. The resulting network is trained and tested on a set of segmented vessels from lung computed tomography scans and then applied to in-vivo photoacoustic measurement data.

show abstract

“…In [23] an error bound for perturbed operators is derived under the classical source condition ∂R(u † ) ∈ range(F [u † ] * ). Under the same source condition, error bounds were also shown in [6,7,33] for more general data fidelity terms, in the first two references for perturbed operators satisfying bounds in Banach lattices. We also refer to [14] for convergence rates of wavelet thresholding methods for perturbed operators.…”

mentioning

confidence: 80%

Error estimates for variational regularization of inverse problems with general noise models for data and operator

Hohage¹,

Werner²

2022

etna

View full text Add to dashboard Cite

This paper is concerned with variational regularization of inverse problems where both the data and the forward operator are given only approximately. We propose a general approach to derive error estimates which separates the analysis of smoothness of the exact solution from the analysis of the effect of errors in the data and the operator. Our abstract error bounds are applied to both discrete and continuous data, random and deterministic types of error, as well as Huber data fidelity terms for impulsive noise.

show abstract

Variational regularisation for inverse problems with imperfect forward operators and general noise models

Cited by 12 publications

References 52 publications

Two-layer neural networks with values in a Banach space

Two-layer neural networks with values in a Banach space

Model-Based Learning for Accelerated, Limited-View 3-D Photoacoustic Tomography

Error estimates for variational regularization of inverse problems with general noise models for data and operator

Contact Info

Product

Resources

About