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  1. Article ; Online: On the approximation of bi-Lipschitz maps by invertible neural networks.

    Jin, Bangti / Zhou, Zehui / Zou, Jun

    Neural networks : the official journal of the International Neural Network Society

    2024  Volume 174, Page(s) 106214

    Abstract: Invertible neural networks (INNs) represent an important class of deep neural network architectures that have been widely used in applications. The universal approximation properties of INNs have been established recently. However, the approximation rate ...

    Abstract Invertible neural networks (INNs) represent an important class of deep neural network architectures that have been widely used in applications. The universal approximation properties of INNs have been established recently. However, the approximation rate of INNs is largely missing. In this work, we provide an analysis of the capacity of a class of coupling-based INNs to approximate bi-Lipschitz continuous mappings on a compact domain, and the result shows that it can well approximate both forward and inverse maps simultaneously. Furthermore, we develop an approach for approximating bi-Lipschitz maps on infinite-dimensional spaces that simultaneously approximate the forward and inverse maps, by combining model reduction with principal component analysis and INNs for approximating the reduced map, and we analyze the overall approximation error of the approach. Preliminary numerical results show the feasibility of the approach for approximating the solution operator for parameterized second-order elliptic problems.
    MeSH term(s) Neural Networks, Computer
    Language English
    Publishing date 2024-02-24
    Publishing country United States
    Document type Journal Article
    ZDB-ID 740542-x
    ISSN 1879-2782 ; 0893-6080
    ISSN (online) 1879-2782
    ISSN 0893-6080
    DOI 10.1016/j.neunet.2024.106214
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    Zs.A 2365: Show issues Location:
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  2. Book ; Online: Electrical Impedance Tomography with Deep Calder\'on Method

    Cen, Siyu / Jin, Bangti / Shin, Kwancheol / Zhou, Zhi

    2023  

    Abstract: Electrical impedance tomography (EIT) is a noninvasive medical imaging modality utilizing the current-density/voltage data measured on the surface of the subject. Calder\'on's method is a relatively recent EIT imaging algorithm that is non-iterative, ... ...

    Abstract Electrical impedance tomography (EIT) is a noninvasive medical imaging modality utilizing the current-density/voltage data measured on the surface of the subject. Calder\'on's method is a relatively recent EIT imaging algorithm that is non-iterative, fast, and capable of reconstructing complex-valued electric impedances. However, due to the regularization via low-pass filtering and linearization, the reconstructed images suffer from severe blurring and under-estimation of the exact conductivity values. In this work, we develop an enhanced version of Calder\'on's method, using {deep} convolution neural networks (i.e., U-net) {as an effective targeted post-processing step, and term the resulting method by deep Calder\'{o}n's method.} Specifically, we learn a U-net to postprocess the EIT images generated by Calder\'on's method so as to have better resolutions and more accurate estimates of conductivity values. We simulate chest configurations with which we generate the current-density/voltage boundary measurements and the corresponding reconstructed images by Calder\'on's method. With the paired training data, we learn the deep neural network and evaluate its performance on real tank measurement data. The experimental results indicate that the proposed approach indeed provides a fast and direct (complex-valued) impedance tomography imaging technique, and substantially improves the capability of the standard Calder\'on's method.

    Comment: 21 pages, appeared at Journal of Computational Physics
    Keywords Mathematics - Numerical Analysis ; Computer Science - Machine Learning
    Subject code 518
    Publishing date 2023-04-18
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  3. Book ; Online: Conductivity Imaging from Internal Measurements with Mixed Least-Squares Deep Neural Networks

    Jin, Bangti / Li, Xiyao / Quan, Qimeng / Zhou, Zhi

    2023  

    Abstract: In this work we develop a novel approach using deep neural networks to reconstruct the conductivity distribution in elliptic problems from one measurement of the solution over the whole domain. The approach is based on a mixed reformulation of the ... ...

    Abstract In this work we develop a novel approach using deep neural networks to reconstruct the conductivity distribution in elliptic problems from one measurement of the solution over the whole domain. The approach is based on a mixed reformulation of the governing equation and utilizes the standard least-squares objective, with deep neural networks as ansatz functions to approximate the conductivity and flux simultaneously. We provide a thorough analysis of the deep neural network approximations of the conductivity for both continuous and empirical losses, including rigorous error estimates that are explicit in terms of the noise level, various penalty parameters and neural network architectural parameters (depth, width and parameter bound). We also provide multiple numerical experiments in two- and multi-dimensions to illustrate distinct features of the approach, e.g., excellent stability with respect to data noise and capability of solving high-dimensional problems.

    Comment: corrected a few typos
    Keywords Mathematics - Numerical Analysis ; Computer Science - Machine Learning
    Subject code 518
    Publishing date 2023-03-29
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  4. Article ; Online: Enhanced reconstruction in magnetic particle imaging by whitening and randomized SVD approximation.

    Kluth, Tobias / Jin, Bangti

    Physics in medicine and biology

    2019  Volume 64, Issue 12, Page(s) 125026

    Abstract: Magnetic particle imaging (MPI) is a medical imaging modality of recent origin, and it exploits the nonlinear magnetization phenomenon to recover a spatially dependent concentration of nanoparticles. In practice, image reconstruction in MPI is frequently ...

    Abstract Magnetic particle imaging (MPI) is a medical imaging modality of recent origin, and it exploits the nonlinear magnetization phenomenon to recover a spatially dependent concentration of nanoparticles. In practice, image reconstruction in MPI is frequently carried out by standard Tikhonov regularization with nonnegativity constraint, which is then minimized by a Kaczmarz type method. In this work, we revisit two issues in the numerical reconstruction in MPI in the lens of inverse theory, i.e. the choice of fidelity and acceleration, and propose two algorithmic tricks, i.e. a whitening procedure to incorporate the noise statistics and accelerating Kaczmarz iteration via randomized SVD. The two tricks are straightforward to implement and easy to incorporate in existing reconstruction algorithms. Their significant potentials are illustrated by extensive numerical experiments on a publicly available dataset.
    MeSH term(s) Algorithms ; Computer Simulation ; Humans ; Image Processing, Computer-Assisted/methods ; Magnetics/instrumentation ; Magnetite Nanoparticles/chemistry ; Molecular Imaging/methods ; Noise ; Phantoms, Imaging
    Chemical Substances Magnetite Nanoparticles
    Language English
    Publishing date 2019-06-21
    Publishing country England
    Document type Journal Article ; Research Support, Non-U.S. Gov't
    ZDB-ID 208857-5
    ISSN 1361-6560 ; 0031-9155
    ISSN (online) 1361-6560
    ISSN 0031-9155
    DOI 10.1088/1361-6560/ab1a4f
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    In stock of ZB MED Cologne/Königswinter

    Zs.A 199: Show issues Location:
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    bis Jg. 1994: Bestellungen von Artikeln über das Online-Bestellformular
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  5. Book ; Online: Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Method

    Hu, Tianhao / Jin, Bangti / Zhou, Zhi

    2022  

    Abstract: In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. This class of problems covers general point sources, line sources and the combination of point-line ...

    Abstract In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. This class of problems covers general point sources, line sources and the combination of point-line sources, and has a broad range of practical applications. The proposed approach is based on decomposing the true solution into a singular part that is known analytically using the fundamental solution of the Laplace equation and a regular part that satisfies a suitable modified elliptic PDE with a smoother source, and then solving for the regular part using the deep Ritz method. A path-following strategy is suggested to select the penalty parameter for enforcing the Dirichlet boundary condition. Extensive numerical experiments in two- and multi-dimensional spaces with point sources, line sources or their combinations are presented to illustrate the efficiency of the proposed approach, and a comparative study with several existing approaches based on neural networks is also given, which shows clearly its competitiveness for the specific class of problems. In addition, we briefly discuss the error analysis of the approach.

    Comment: 28 pages
    Keywords Mathematics - Numerical Analysis ; Computer Science - Machine Learning
    Subject code 518
    Publishing date 2022-09-07
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  6. Book ; Online: Imaging Conductivity from Current Density Magnitude using Neural Networks

    Jin, Bangti / Li, Xiyao / Lu, Xiliang

    2022  

    Abstract: Conductivity imaging represents one of the most important tasks in medical imaging. In this work we develop a neural network based reconstruction technique for imaging the conductivity from the magnitude of the internal current density. It is achieved by ...

    Abstract Conductivity imaging represents one of the most important tasks in medical imaging. In this work we develop a neural network based reconstruction technique for imaging the conductivity from the magnitude of the internal current density. It is achieved by formulating the problem as a relaxed weighted least-gradient problem, and then approximating its minimizer by standard fully connected feedforward neural networks. We derive bounds on two components of the generalization error, i.e., approximation error and statistical error, explicitly in terms of properties of the neural networks (e.g., depth, total number of parameters, and the bound of the network parameters). We illustrate the performance and distinct features of the approach on several numerical experiments. Numerically, it is observed that the approach enjoys remarkable robustness with respect to the presence of data noise.

    Comment: 29 pp, 9 figures (several typos are corrected in the new version), to appear at Inverse Problems
    Keywords Mathematics - Numerical Analysis ; Computer Science - Machine Learning ; Electrical Engineering and Systems Science - Image and Video Processing
    Subject code 006
    Publishing date 2022-04-05
    Publishing country us
    Document type Book ; Online
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  7. Book ; Online: L1 data fitting for robust reconstruction in magnetic particle imaging

    Kluth, Tobias / Jin, Bangti

    quantitative evaluation on Open MPI dataset

    2020  

    Abstract: Magnetic particle imaging is an emerging quantitative imaging modality, exploiting the unique nonlinear magnetization phenomenon of superparamagnetic iron oxide nanoparticles for recovering the concentration. Traditionally the reconstruction is ... ...

    Abstract Magnetic particle imaging is an emerging quantitative imaging modality, exploiting the unique nonlinear magnetization phenomenon of superparamagnetic iron oxide nanoparticles for recovering the concentration. Traditionally the reconstruction is formulated into a penalized least-squares problem with nonnegativity constraint, and then solved using a variant of Kaczmarz method which is often stopped early after a small number of iterations. Besides the phantom signal, measurements additionally include a background signal and a noise signal. In order to obtain good reconstructions, a preprocessing step of frequency selection to remove the deleterious influences of the noise is often adopted. In this work, we propose a complementary pure variational approach to noise treatment, by viewing highly noisy measurements as outliers, and employing the l1 data fitting, one popular approach from robust statistics. When compared with the standard approach, it is easy to implement with a comparable computational complexity. Experiments with a public domain dataset, i.e., Open MPI dataset, show that it can give accurate reconstructions, and is less prone to noisy measurements, which is illustrated by quantitative (PSNR / SSIM) and qualitative comparisons with the Kaczmarz method. We also investigate the performance of the Kaczmarz method for small iteration numbers quantitatively.
    Keywords Mathematics - Numerical Analysis ; Electrical Engineering and Systems Science - Image and Video Processing
    Subject code 620
    Publishing date 2020-01-16
    Publishing country us
    Document type Book ; Online
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  8. Article: Unsupervised knowledge-transfer for learned image reconstruction.

    Barbano, Riccardo / Kereta, Željko / Hauptmann, Andreas / Arridge, Simon R / Jin, Bangti

    Inverse problems

    2022  Volume 38, Issue 10, Page(s) 104004

    Abstract: Deep learning-based image reconstruction approaches have demonstrated impressive empirical performance in many imaging modalities. These approaches usually require a large amount of high-quality paired training data, which is often not available in ... ...

    Abstract Deep learning-based image reconstruction approaches have demonstrated impressive empirical performance in many imaging modalities. These approaches usually require a large amount of high-quality paired training data, which is often not available in medical imaging. To circumvent this issue we develop a novel unsupervised knowledge-transfer paradigm for learned reconstruction within a Bayesian framework. The proposed approach learns a reconstruction network in two phases. The first phase trains a reconstruction network with a set of ordered pairs comprising of ground truth images of ellipses and the corresponding simulated measurement data. The second phase fine-tunes the pretrained network to more realistic measurement data without supervision. By construction, the framework is capable of delivering predictive uncertainty information over the reconstructed image. We present extensive experimental results on low-dose and sparse-view computed tomography showing that the approach is competitive with several state-of-the-art supervised and unsupervised reconstruction techniques. Moreover, for test data distributed differently from the training data, the proposed framework can significantly improve reconstruction quality not only visually, but also quantitatively in terms of PSNR and SSIM, when compared with learned methods trained on the synthetic dataset only.
    Language English
    Publishing date 2022-09-08
    Publishing country England
    Document type Journal Article
    ZDB-ID 1477292-9
    ISSN 1361-6420 ; 0266-5611
    ISSN (online) 1361-6420
    ISSN 0266-5611
    DOI 10.1088/1361-6420/ac8a91
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  9. Book ; Online: Probabilistic Residual Learning for Aleatoric Uncertainty in Image Restoration

    Zhang, Chen / Jin, Bangti

    2019  

    Abstract: Aleatoric uncertainty is an intrinsic property of ill-posed inverse and imaging problems. Its quantification is vital for assessing the reliability of relevant point estimates. In this paper, we propose an efficient framework for quantifying aleatoric ... ...

    Abstract Aleatoric uncertainty is an intrinsic property of ill-posed inverse and imaging problems. Its quantification is vital for assessing the reliability of relevant point estimates. In this paper, we propose an efficient framework for quantifying aleatoric uncertainty for deep residual learning and showcase its significant potential on image restoration. In the framework, we divide the conditional probability modeling for the residual variable into a deterministic homo-dimensional level, a stochastic low-dimensional level and a merging level. The low-dimensionality is especially suitable for sparse correlation between image pixels, enables efficient sampling for high dimensional problems and acts as a regularizer for the distribution. Preliminary numerical experiments show that the proposed method can give not only state-of-the-art point estimates of image restoration but also useful associated uncertainty information.

    Comment: this version is outdated, and we are completely reorganizing the paper and split it into several different pieces of work. Thus, we prefer to withdraw it from arxiv
    Keywords Electrical Engineering and Systems Science - Image and Video Processing ; Computer Science - Machine Learning ; Statistics - Machine Learning
    Subject code 006
    Publishing date 2019-08-01
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  10. Book ; Online: Conditional Variational Autoencoder for Learned Image Reconstruction

    Zhang, Chen / Barbano, Riccardo / Jin, Bangti

    2021  

    Abstract: Learned image reconstruction techniques using deep neural networks have recently gained popularity, and have delivered promising empirical results. However, most approaches focus on one single recovery for each observation, and thus neglect the ... ...

    Abstract Learned image reconstruction techniques using deep neural networks have recently gained popularity, and have delivered promising empirical results. However, most approaches focus on one single recovery for each observation, and thus neglect the uncertainty information. In this work, we develop a novel computational framework that approximates the posterior distribution of the unknown image at each query observation. The proposed framework is very flexible: It handles implicit noise models and priors, it incorporates the data formation process (i.e., the forward operator), and the learned reconstructive properties are transferable between different datasets. Once the network is trained using the conditional variational autoencoder loss, it provides a computationally efficient sampler for the approximate posterior distribution via feed-forward propagation, and the summarizing statistics of the generated samples are used for both point-estimation and uncertainty quantification. We illustrate the proposed framework with extensive numerical experiments on positron emission tomography (with both moderate and low count levels) showing that the framework generates high-quality samples when compared with state-of-the-art methods.

    Comment: 22 pages, preliminary version appeared as 1908.01010
    Keywords Computer Science - Computer Vision and Pattern Recognition ; Mathematics - Numerical Analysis
    Subject code 006
    Publishing date 2021-10-22
    Publishing country us
    Document type Book ; Online
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