SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 147-187, March 2024. Abstract. In this work, we develop a novel approach using deep neural networks (DNNs) 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 DNNs as ansatz functions to approximate the conductivity and flux simultaneously. We provide a thorough analysis of the DNN 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 bounds). We also provide multiple numerical experiments in two dimensions and multidimensions to illustrate distinct features of the approach, e.g., excellent stability with respect to data noise and capability of solving high-dimensional problems.
{"title":"Conductivity Imaging from Internal Measurements with Mixed Least-Squares Deep Neural Networks","authors":"Bangti Jin, Xiyao Li, Qimeng Quan, Zhi Zhou","doi":"10.1137/23m1562536","DOIUrl":"https://doi.org/10.1137/23m1562536","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 147-187, March 2024. <br/> Abstract. In this work, we develop a novel approach using deep neural networks (DNNs) 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 DNNs as ansatz functions to approximate the conductivity and flux simultaneously. We provide a thorough analysis of the DNN 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 bounds). We also provide multiple numerical experiments in two dimensions and multidimensions to illustrate distinct features of the approach, e.g., excellent stability with respect to data noise and capability of solving high-dimensional problems.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"255 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Siddharth S. Iyer, Frank Ong, Xiaozhi Cao, Congyu Liao, Luca Daniel, Jonathan I. Tamir, Kawin Setsompop
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 116-146, March 2024. Abstract. This work aims to accelerate the convergence of proximal gradient methods used to solve regularized linear inverse problems. This is achieved by designing a polynomial-based preconditioner that targets the eigenvalue spectrum of the normal operator derived from the linear operator. The preconditioner does not assume any explicit structure on the linear function and thus can be deployed in diverse applications of interest. The efficacy of the preconditioner is validated on three different Magnetic Resonance Imaging applications, where it is seen to achieve faster iterative convergence (around [math] faster, depending on the application of interest) while achieving similar reconstruction quality.
{"title":"Polynomial Preconditioners for Regularized Linear Inverse Problems","authors":"Siddharth S. Iyer, Frank Ong, Xiaozhi Cao, Congyu Liao, Luca Daniel, Jonathan I. Tamir, Kawin Setsompop","doi":"10.1137/22m1530355","DOIUrl":"https://doi.org/10.1137/22m1530355","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 116-146, March 2024. <br/> Abstract. This work aims to accelerate the convergence of proximal gradient methods used to solve regularized linear inverse problems. This is achieved by designing a polynomial-based preconditioner that targets the eigenvalue spectrum of the normal operator derived from the linear operator. The preconditioner does not assume any explicit structure on the linear function and thus can be deployed in diverse applications of interest. The efficacy of the preconditioner is validated on three different Magnetic Resonance Imaging applications, where it is seen to achieve faster iterative convergence (around [math] faster, depending on the application of interest) while achieving similar reconstruction quality.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"211 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139516023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 91-115, March 2024. Abstract.We propose to learn non-convex regularizers with a prescribed upper bound on their weak-convexity modulus. Such regularizers give rise to variational denoisers that minimize a convex energy. They rely on few parameters (less than 15,000) and offer a signal-processing interpretation as they mimic handcrafted sparsity-promoting regularizers. Through numerical experiments, we show that such denoisers outperform convex-regularization methods as well as the popular BM3D denoiser. Additionally, the learned regularizer can be deployed to solve inverse problems with iterative schemes that provably converge. For both CT and MRI reconstruction, the regularizer generalizes well and offers an excellent tradeoff between performance, number of parameters, guarantees, and interpretability when compared to other data-driven approaches.
{"title":"Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms","authors":"Alexis Goujon, Sebastian Neumayer, Michael Unser","doi":"10.1137/23m1565243","DOIUrl":"https://doi.org/10.1137/23m1565243","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 91-115, March 2024. <br/> Abstract.We propose to learn non-convex regularizers with a prescribed upper bound on their weak-convexity modulus. Such regularizers give rise to variational denoisers that minimize a convex energy. They rely on few parameters (less than 15,000) and offer a signal-processing interpretation as they mimic handcrafted sparsity-promoting regularizers. Through numerical experiments, we show that such denoisers outperform convex-regularization methods as well as the popular BM3D denoiser. Additionally, the learned regularizer can be deployed to solve inverse problems with iterative schemes that provably converge. For both CT and MRI reconstruction, the regularizer generalizes well and offers an excellent tradeoff between performance, number of parameters, guarantees, and interpretability when compared to other data-driven approaches.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"266 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139498043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 61-90, March 2024. Abstract. We consider the task of estimating the unknown diffusion parameter in an elliptic PDE as a model problem to develop and test the effectiveness and robustness to noise of reconstruction schemes with sparsity regularization. To this end, the model problem is recast as a nonlinear infinite dimensional optimization problem, where the logarithm of the unknown diffusion parameter is modeled using a linear combination of the elements of a dictionary, i.e., a known bounded sequence of [math] functions, with unknown coefficients that form a sequence in [math]. We show that the regularization of this nonlinear optimization problem using a weighted [math]-norm has minimizers that are finitely supported. We then propose modifications of well-known algorithms (ISTA and FISTA) to find a minimizer of this weighted [math]-norm regularized nonlinear optimization problem that accounts for the fact that in general the smooth part of the functional being optimized is a functional only defined over [math]. We also introduce semismooth methods (ASISTA and FASISTA) for finding a minimizer, which locally uses Gauss–Newton type surrogate models that additionally are stabilized by means of a Levenberg–Marquardt type approach. Our numerical examples show that the regularization with the weighted [math]-norm indeed does make the estimation more robust with respect to noise. Moreover, the numerical examples also demonstrate that the ASISTA and FASISTA methods are quite efficient, outperforming both ISTA and FISTA.
{"title":"Identification of Sparsely Representable Diffusion Parameters in Elliptic Problems","authors":"Luzia N. Felber, Helmut Harbrecht, Marc Schmidlin","doi":"10.1137/23m1565346","DOIUrl":"https://doi.org/10.1137/23m1565346","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 61-90, March 2024. <br/> Abstract. We consider the task of estimating the unknown diffusion parameter in an elliptic PDE as a model problem to develop and test the effectiveness and robustness to noise of reconstruction schemes with sparsity regularization. To this end, the model problem is recast as a nonlinear infinite dimensional optimization problem, where the logarithm of the unknown diffusion parameter is modeled using a linear combination of the elements of a dictionary, i.e., a known bounded sequence of [math] functions, with unknown coefficients that form a sequence in [math]. We show that the regularization of this nonlinear optimization problem using a weighted [math]-norm has minimizers that are finitely supported. We then propose modifications of well-known algorithms (ISTA and FISTA) to find a minimizer of this weighted [math]-norm regularized nonlinear optimization problem that accounts for the fact that in general the smooth part of the functional being optimized is a functional only defined over [math]. We also introduce semismooth methods (ASISTA and FASISTA) for finding a minimizer, which locally uses Gauss–Newton type surrogate models that additionally are stabilized by means of a Levenberg–Marquardt type approach. Our numerical examples show that the regularization with the weighted [math]-norm indeed does make the estimation more robust with respect to noise. Moreover, the numerical examples also demonstrate that the ASISTA and FASISTA methods are quite efficient, outperforming both ISTA and FISTA.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139498035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Avrajit Ghosh, Michael McCann, Madeline Mitchell, Saiprasad Ravishankar
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 31-60, March 2024. Abstract. We present a gradient-based heuristic method for supervised learning of sparsity-promoting regularizers for denoising signals and images. Sparsity-promoting regularization is a key ingredient in solving modern signal reconstruction problems; however, the operators underlying these regularizers are usually either designed by hand or learned from data in an unsupervised way. The recent success of supervised learning (e.g., with convolutional neural networks) in solving image reconstruction problems suggests that it could be a fruitful approach to designing regularizers. Towards this end, we propose to denoise signals using a variational formulation with a parametric, sparsity-promoting regularizer, where the parameters of the regularizer are learned to minimize the mean squared error of reconstructions on a training set of ground truth image and measurement pairs. Training involves solving a challenging bilevel optimization problem; we derive an expression for the gradient of the training loss using the closed-form solution of the denoising problem and provide an accompanying gradient descent algorithm to minimize it. Our experiments with structured 1D signals and natural images indicate that the proposed method can learn an operator that outperforms well-known regularizers (total variation, DCT-sparsity, and unsupervised dictionary learning) and collaborative filtering for denoising.
{"title":"Learning Sparsity-Promoting Regularizers Using Bilevel Optimization","authors":"Avrajit Ghosh, Michael McCann, Madeline Mitchell, Saiprasad Ravishankar","doi":"10.1137/22m1506547","DOIUrl":"https://doi.org/10.1137/22m1506547","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 31-60, March 2024. <br/> Abstract. We present a gradient-based heuristic method for supervised learning of sparsity-promoting regularizers for denoising signals and images. Sparsity-promoting regularization is a key ingredient in solving modern signal reconstruction problems; however, the operators underlying these regularizers are usually either designed by hand or learned from data in an unsupervised way. The recent success of supervised learning (e.g., with convolutional neural networks) in solving image reconstruction problems suggests that it could be a fruitful approach to designing regularizers. Towards this end, we propose to denoise signals using a variational formulation with a parametric, sparsity-promoting regularizer, where the parameters of the regularizer are learned to minimize the mean squared error of reconstructions on a training set of ground truth image and measurement pairs. Training involves solving a challenging bilevel optimization problem; we derive an expression for the gradient of the training loss using the closed-form solution of the denoising problem and provide an accompanying gradient descent algorithm to minimize it. Our experiments with structured 1D signals and natural images indicate that the proposed method can learn an operator that outperforms well-known regularizers (total variation, DCT-sparsity, and unsupervised dictionary learning) and collaborative filtering for denoising.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"180 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139411114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 1-30, March 2024. Abstract. Low-light image enhancement plays an important role in computer vision applications, which is a fundamental low-level task and can affect high-level computer vision tasks. To solve this ill-posed problem, a lot of methods have been proposed to enhance low-light images. However, their performance degrades significantly under nonuniform lighting conditions. Due to the rapid variation of illuminance in different regions in natural images, it is challenging to enhance low-light parts and retain normal-light parts simultaneously in the same image. Commonly, either the low-light parts are underenhanced or the normal-light parts are overenhanced, accompanied by color distortion and artifacts. To overcome this problem, we propose a simple and effective Retinex-based model with reflectance map reweighting for images under nonuniform lighting conditions. An alternating proximal gradient (APG) algorithm is proposed to solve the proposed model, in which the illumination map, the reflectance map, and the weighting map are updated iteratively. To make our model applicable to a wide range of light conditions, we design an initialization scheme for the weighting map. A theoretical analysis of the existence of the solution to our model and the convergence of the APG algorithm are also established. A series of experiments on real-world low-light images are conducted, which demonstrate the effectiveness of our method.
{"title":"A Variational Model for Nonuniform Low-Light Image Enhancement","authors":"Fan Jia, Shen Mao, Xue-Cheng Tai, Tieyong Zeng","doi":"10.1137/22m1543161","DOIUrl":"https://doi.org/10.1137/22m1543161","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 1-30, March 2024. <br/> Abstract. Low-light image enhancement plays an important role in computer vision applications, which is a fundamental low-level task and can affect high-level computer vision tasks. To solve this ill-posed problem, a lot of methods have been proposed to enhance low-light images. However, their performance degrades significantly under nonuniform lighting conditions. Due to the rapid variation of illuminance in different regions in natural images, it is challenging to enhance low-light parts and retain normal-light parts simultaneously in the same image. Commonly, either the low-light parts are underenhanced or the normal-light parts are overenhanced, accompanied by color distortion and artifacts. To overcome this problem, we propose a simple and effective Retinex-based model with reflectance map reweighting for images under nonuniform lighting conditions. An alternating proximal gradient (APG) algorithm is proposed to solve the proposed model, in which the illumination map, the reflectance map, and the weighting map are updated iteratively. To make our model applicable to a wide range of light conditions, we design an initialization scheme for the weighting map. A theoretical analysis of the existence of the solution to our model and the convergence of the APG algorithm are also established. A series of experiments on real-world low-light images are conducted, which demonstrate the effectiveness of our method.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"60 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 16, Issue 4, Page 2247-2284, December 2023. Abstract. Deep learning has proved to be a powerful tool for solving inverse problems in imaging, and most of the related work is based on supervised learning. In many applications, collecting truth images is a challenging and costly task, and the prerequisite of having a training dataset of truth images limits its applicability. This paper proposes a self-supervised deep learning method for solving inverse imaging problems that does not require any training samples. The proposed approach is built on a reparametrization of latent images using a convolutional neural network, and the reconstruction is motivated by approximating the minimum mean square error estimate of the latent image using a Langevin dynamics–based Monte Carlo (MC) method. To efficiently sample the network weights in the context of image reconstruction, we propose a Langevin MC scheme called Adam-LD, inspired by the well-known optimizer in deep learning, Adam. The proposed method is applied to solve linear and nonlinear inverse problems, specifically, sparse-view computed tomography image reconstruction and phase retrieval. Our experiments demonstrate that the proposed method outperforms existing unsupervised or self-supervised solutions in terms of reconstruction quality.
{"title":"Self-Supervised Deep Learning for Image Reconstruction: A Langevin Monte Carlo Approach","authors":"Ji Li, Weixi Wang, Hui Ji","doi":"10.1137/23m1548025","DOIUrl":"https://doi.org/10.1137/23m1548025","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 16, Issue 4, Page 2247-2284, December 2023. <br/> Abstract. Deep learning has proved to be a powerful tool for solving inverse problems in imaging, and most of the related work is based on supervised learning. In many applications, collecting truth images is a challenging and costly task, and the prerequisite of having a training dataset of truth images limits its applicability. This paper proposes a self-supervised deep learning method for solving inverse imaging problems that does not require any training samples. The proposed approach is built on a reparametrization of latent images using a convolutional neural network, and the reconstruction is motivated by approximating the minimum mean square error estimate of the latent image using a Langevin dynamics–based Monte Carlo (MC) method. To efficiently sample the network weights in the context of image reconstruction, we propose a Langevin MC scheme called Adam-LD, inspired by the well-known optimizer in deep learning, Adam. The proposed method is applied to solve linear and nonlinear inverse problems, specifically, sparse-view computed tomography image reconstruction and phase retrieval. Our experiments demonstrate that the proposed method outperforms existing unsupervised or self-supervised solutions in terms of reconstruction quality.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"33 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138528994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Andreas Kofler, Fabian Altekrüger, Fatima Antarou Ba, Christoph Kolbitsch, Evangelos Papoutsellis, David Schote, Clemens Sirotenko, Felix Frederik Zimmermann, Kostas Papafitsoros
SIAM Journal on Imaging Sciences, Volume 16, Issue 4, Page 2202-2246, December 2023. Abstract. We introduce a method for the fast estimation of data-adapted, spatially and temporally dependent regularization parameter-maps for variational image reconstruction, focusing on total variation (TV) minimization. The proposed approach is inspired by recent developments in algorithm unrolling using deep neural networks (NNs) and relies on two distinct subnetworks. The first subnetwork estimates the regularization parameter-map from the input data. The second subnetwork unrolls [math] iterations of an iterative algorithm which approximately solves the corresponding TV-minimization problem incorporating the previously estimated regularization parameter-map. The overall network is then trained end-to-end in a supervised learning fashion using pairs of clean and corrupted data but crucially without the need for access to labels for the optimal regularization parameter-maps. We first prove consistency of the unrolled scheme by showing that the unrolled minimizing energy functional used for the supervised learning [math]-converges, as [math] tends to infinity, to the corresponding functional that incorporates the exact solution map of the TV-minimization problem. Then, we apply and evaluate the proposed method on a variety of large-scale and dynamic imaging problems with retrospectively simulated measurement data for which the automatic computation of such regularization parameters has been so far challenging using the state-of-the-art methods: a 2D dynamic cardiac magnetic resonance imaging (MRI) reconstruction problem, a quantitative brain MRI reconstruction problem, a low-dose computed tomography problem, and a dynamic image denoising problem. The proposed method consistently improves the TV reconstructions using scalar regularization parameters, and the obtained regularization parameter-maps adapt well to imaging problems and data by leading to the preservation of detailed features. Although the choice of the regularization parameter-maps is data-driven and based on NNs, the subsequent reconstruction algorithm is interpretable since it inherits the properties (e.g., convergence guarantees) of the iterative reconstruction method from which the network is implicitly defined.
{"title":"Learning Regularization Parameter-Maps for Variational Image Reconstruction Using Deep Neural Networks and Algorithm Unrolling","authors":"Andreas Kofler, Fabian Altekrüger, Fatima Antarou Ba, Christoph Kolbitsch, Evangelos Papoutsellis, David Schote, Clemens Sirotenko, Felix Frederik Zimmermann, Kostas Papafitsoros","doi":"10.1137/23m1552486","DOIUrl":"https://doi.org/10.1137/23m1552486","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 16, Issue 4, Page 2202-2246, December 2023. <br/> Abstract. We introduce a method for the fast estimation of data-adapted, spatially and temporally dependent regularization parameter-maps for variational image reconstruction, focusing on total variation (TV) minimization. The proposed approach is inspired by recent developments in algorithm unrolling using deep neural networks (NNs) and relies on two distinct subnetworks. The first subnetwork estimates the regularization parameter-map from the input data. The second subnetwork unrolls [math] iterations of an iterative algorithm which approximately solves the corresponding TV-minimization problem incorporating the previously estimated regularization parameter-map. The overall network is then trained end-to-end in a supervised learning fashion using pairs of clean and corrupted data but crucially without the need for access to labels for the optimal regularization parameter-maps. We first prove consistency of the unrolled scheme by showing that the unrolled minimizing energy functional used for the supervised learning [math]-converges, as [math] tends to infinity, to the corresponding functional that incorporates the exact solution map of the TV-minimization problem. Then, we apply and evaluate the proposed method on a variety of large-scale and dynamic imaging problems with retrospectively simulated measurement data for which the automatic computation of such regularization parameters has been so far challenging using the state-of-the-art methods: a 2D dynamic cardiac magnetic resonance imaging (MRI) reconstruction problem, a quantitative brain MRI reconstruction problem, a low-dose computed tomography problem, and a dynamic image denoising problem. The proposed method consistently improves the TV reconstructions using scalar regularization parameters, and the obtained regularization parameter-maps adapt well to imaging problems and data by leading to the preservation of detailed features. Although the choice of the regularization parameter-maps is data-driven and based on NNs, the subsequent reconstruction algorithm is interpretable since it inherits the properties (e.g., convergence guarantees) of the iterative reconstruction method from which the network is implicitly defined.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"69 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 16, Issue 4, Page 2175-2201, December 2023. Abstract. We consider the problem of reconstructing one-dimensional point sources from their Fourier measurements in a bounded interval [math]. This problem is known to be challenging in the regime where the spacing of the sources is below the Rayleigh length [math]. In this paper, we propose a superresolution algorithm, called iterative focusing-localization and iltering, to resolve closely spaced point sources from their multiple measurements that are obtained by using multiple unknown illumination patterns. The new proposed algorithm has a distinct feature in that it reconstructs the point sources one by one in an iterative manner and hence requires no prior information about the source numbers. The new feature also allows for a subsampling strategy that can reconstruct sources using small-sized Hankel matrices and thus circumvent the computation of singular-value decomposition for large matrices as in the usual subspace methods. In addition, the algorithm can be paralleled. A theoretical analysis of the methods behind the algorithm is also provided. The derived results imply a phase transition phenomenon in the reconstruction of source locations which is confirmed in the numerical experiment. Numerical results show that the algorithm can achieve a stable reconstruction for point sources with a minimum separation distance that is close to the theoretical limit. The efficiency and robustness of the algorithm have also been tested. This algorithm can be generalized to higher dimensions.
{"title":"IFF: A Superresolution Algorithm for Multiple Measurements","authors":"Zetao Fei, Hai Zhang","doi":"10.1137/23m1568569","DOIUrl":"https://doi.org/10.1137/23m1568569","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 16, Issue 4, Page 2175-2201, December 2023. <br/> Abstract. We consider the problem of reconstructing one-dimensional point sources from their Fourier measurements in a bounded interval [math]. This problem is known to be challenging in the regime where the spacing of the sources is below the Rayleigh length [math]. In this paper, we propose a superresolution algorithm, called iterative focusing-localization and iltering, to resolve closely spaced point sources from their multiple measurements that are obtained by using multiple unknown illumination patterns. The new proposed algorithm has a distinct feature in that it reconstructs the point sources one by one in an iterative manner and hence requires no prior information about the source numbers. The new feature also allows for a subsampling strategy that can reconstruct sources using small-sized Hankel matrices and thus circumvent the computation of singular-value decomposition for large matrices as in the usual subspace methods. In addition, the algorithm can be paralleled. A theoretical analysis of the methods behind the algorithm is also provided. The derived results imply a phase transition phenomenon in the reconstruction of source locations which is confirmed in the numerical experiment. Numerical results show that the algorithm can achieve a stable reconstruction for point sources with a minimum separation distance that is close to the theoretical limit. The efficiency and robustness of the algorithm have also been tested. This algorithm can be generalized to higher dimensions.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"25 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138528995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Imaging Sciences, Volume 16, Issue 4, Page 2144-2174, December 2023. Abstract. Turbulent fluctuations of the electron number density in the Earth’s ionosphere may hamper the performance of spaceborne synthetic aperture radar (SAR). Previously, we have quantified the extent of the possible degradation of transionospheric SAR images as it depends on the state of the ionosphere and parameters of the SAR instrument. Yet no attempt has been made to mitigate the adverse effect of the ionospheric turbulence. In the current work, we propose a new optimization-based autofocus algorithm that helps correct the turbulence-induced distortions of spaceborne SAR images. Unlike the traditional autofocus procedures available in the literature, the new algorithm allows for the dependence of the phase perturbations of SAR signals not only on slow time but also on the target coordinates. This dependence is central for the analysis of image distortions due to turbulence, but in the case of traditional autofocus where the distortions are due to uncertainties in the antenna position, it is not present.
{"title":"Transionospheric Autofocus for Synthetic Aperture Radar","authors":"Mikhail Gilman, Semyon V. Tsynkov","doi":"10.1137/22m153570x","DOIUrl":"https://doi.org/10.1137/22m153570x","url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 16, Issue 4, Page 2144-2174, December 2023. <br/> Abstract. Turbulent fluctuations of the electron number density in the Earth’s ionosphere may hamper the performance of spaceborne synthetic aperture radar (SAR). Previously, we have quantified the extent of the possible degradation of transionospheric SAR images as it depends on the state of the ionosphere and parameters of the SAR instrument. Yet no attempt has been made to mitigate the adverse effect of the ionospheric turbulence. In the current work, we propose a new optimization-based autofocus algorithm that helps correct the turbulence-induced distortions of spaceborne SAR images. Unlike the traditional autofocus procedures available in the literature, the new algorithm allows for the dependence of the phase perturbations of SAR signals not only on slow time but also on the target coordinates. This dependence is central for the analysis of image distortions due to turbulence, but in the case of traditional autofocus where the distortions are due to uncertainties in the antenna position, it is not present.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138528999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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