The nonconvex optimization method has attracted increasing attention due to its excellent ability of promoting sparsity in signal processing, image restoration, and machine learning. In this paper, we consider a new minimization method and its applications in signal recovery and image reconstruction because minimization provides an effective way to solve the -ratio sparsity minimization model. Our main contributions are to establish a convex hull decomposition for and investigate RIP-based conditions for stable signal recovery and image reconstruction by minimization. For one-dimensional signal recovery, our derived RIP condition extends existing results. For two-dimensional image recovery under minimization of image gradients, we provide the error estimate of the resulting optimal solutions in terms of sparsity and noise level, which is missing in the literature. Numerical results of the limited angle problem in computed tomography imaging and image deblurring are presented to validate the efficiency and superiority of the proposed minimization method among the state-of-art image recovery methods.
{"title":"(boldsymbol{L_1-beta L_q}) Minimization for Signal and Image Recovery","authors":"Limei Huo, Wengu Chen, Huanmin Ge, Michael K. Ng","doi":"10.1137/22m1525363","DOIUrl":"https://doi.org/10.1137/22m1525363","url":null,"abstract":"The nonconvex optimization method has attracted increasing attention due to its excellent ability of promoting sparsity in signal processing, image restoration, and machine learning. In this paper, we consider a new minimization method and its applications in signal recovery and image reconstruction because minimization provides an effective way to solve the -ratio sparsity minimization model. Our main contributions are to establish a convex hull decomposition for and investigate RIP-based conditions for stable signal recovery and image reconstruction by minimization. For one-dimensional signal recovery, our derived RIP condition extends existing results. For two-dimensional image recovery under minimization of image gradients, we provide the error estimate of the resulting optimal solutions in terms of sparsity and noise level, which is missing in the literature. Numerical results of the limited angle problem in computed tomography imaging and image deblurring are presented to validate the efficiency and superiority of the proposed minimization method among the state-of-art image recovery methods.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136210799","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}
Rima Alaifari, Giovanni S. Alberti, Tandri Gauksson
{"title":"Short Communication: Localized Adversarial Artifacts for Compressed Sensing MRI","authors":"Rima Alaifari, Giovanni S. Alberti, Tandri Gauksson","doi":"10.1137/22m1503221","DOIUrl":"https://doi.org/10.1137/22m1503221","url":null,"abstract":"","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136295947","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}
{"title":"Convergence Analysis of Volumetric Stretch Energy Minimization and Its Associated Optimal Mass Transport","authors":"Tsung-Ming Huang, Wei-Hung Liao, Wen-Wei Lin, Mei-Heng Yueh, Shing-Tung Yau","doi":"10.1137/22m1528756","DOIUrl":"https://doi.org/10.1137/22m1528756","url":null,"abstract":"","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136362298","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}
Michael V. Klibanov, Jingzhi Li, Loc H. Nguyen, Vladimir Romanov, Zhipeng Yang
The first globally convergent numerical method for a coefficient inverse problem for the Riemannian radiative transfer equation (RRTE) is constructed. This is a version of the so-called convexification method, which has been pursued by this research group for a number of years for some other CIPs for PDEs. Those PDEs are significantly different from the RRTE. The presence of the Carleman weight function in the numerical scheme is the key element which insures the global convergence. Convergence analysis is presented along with the results of numerical experiments, which confirm the theory. RRTE governs the propagation of photons in the diffuse medium in the case when they propagate along geodesic lines between their collisions. Geodesic lines are generated by the spatially variable dielectric constant of the medium.
{"title":"Convexification Numerical Method for a Coefficient Inverse Problem for the Riemannian Radiative Transfer Equation","authors":"Michael V. Klibanov, Jingzhi Li, Loc H. Nguyen, Vladimir Romanov, Zhipeng Yang","doi":"10.1137/23m1565449","DOIUrl":"https://doi.org/10.1137/23m1565449","url":null,"abstract":"The first globally convergent numerical method for a coefficient inverse problem for the Riemannian radiative transfer equation (RRTE) is constructed. This is a version of the so-called convexification method, which has been pursued by this research group for a number of years for some other CIPs for PDEs. Those PDEs are significantly different from the RRTE. The presence of the Carleman weight function in the numerical scheme is the key element which insures the global convergence. Convergence analysis is presented along with the results of numerical experiments, which confirm the theory. RRTE governs the propagation of photons in the diffuse medium in the case when they propagate along geodesic lines between their collisions. Geodesic lines are generated by the spatially variable dielectric constant of the medium.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136242991","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}
Image denoising—removal of additive white Gaussian noise from an image—is one of the oldest and most studied problems in image processing. Extensive work over several decades has led to thousands of papers on this subject, and to many well-performing algorithms for this task. Indeed, 10 years ago, these achievements led some researchers to suspect that “Denoising is Dead,” in the sense that all that can be achieved in this domain has already been obtained. However, this turned out to be far from the truth, with the penetration of deep learning (DL) into the realm of image processing. The era of DL brought a revolution to image denoising, both by taking the lead in today’s ability for noise suppression in images, and by broadening the scope of denoising problems being treated. Our paper starts by describing this evolution, highlighting in particular the tension and synergy that exist between classical approaches and modern artificial intelligence (AI) alternatives in design of image denoisers. The recent transitions in the field of image denoising go far beyond the ability to design better denoisers. In the second part of this paper we focus on recently discovered abilities and prospects of image denoisers. We expose the possibility of using image denoisers for service of other problems, such as regularizing general inverse problems and serving as the prime engine in diffusion-based image synthesis. We also unveil the (strange?) idea that denoising and other inverse problems might not have a unique solution, as common algorithms would have us believe. Instead, we describe constructive ways to produce randomized and diverse high perceptual quality results for inverse problems, all fueled by the progress that DL brought to image denoising. This is a survey paper, and its prime goal is to provide a broad view of the history of the field of image denoising and closely related topics in image processing. Our aim is to give a better context to recent discoveries, and to the influence of the AI revolution in our domain.
{"title":"Image Denoising: The Deep Learning Revolution and Beyond—A Survey Paper","authors":"Michael Elad, Bahjat Kawar, Gregory Vaksman","doi":"10.1137/23m1545859","DOIUrl":"https://doi.org/10.1137/23m1545859","url":null,"abstract":"Image denoising—removal of additive white Gaussian noise from an image—is one of the oldest and most studied problems in image processing. Extensive work over several decades has led to thousands of papers on this subject, and to many well-performing algorithms for this task. Indeed, 10 years ago, these achievements led some researchers to suspect that “Denoising is Dead,” in the sense that all that can be achieved in this domain has already been obtained. However, this turned out to be far from the truth, with the penetration of deep learning (DL) into the realm of image processing. The era of DL brought a revolution to image denoising, both by taking the lead in today’s ability for noise suppression in images, and by broadening the scope of denoising problems being treated. Our paper starts by describing this evolution, highlighting in particular the tension and synergy that exist between classical approaches and modern artificial intelligence (AI) alternatives in design of image denoisers. The recent transitions in the field of image denoising go far beyond the ability to design better denoisers. In the second part of this paper we focus on recently discovered abilities and prospects of image denoisers. We expose the possibility of using image denoisers for service of other problems, such as regularizing general inverse problems and serving as the prime engine in diffusion-based image synthesis. We also unveil the (strange?) idea that denoising and other inverse problems might not have a unique solution, as common algorithms would have us believe. Instead, we describe constructive ways to produce randomized and diverse high perceptual quality results for inverse problems, all fueled by the progress that DL brought to image denoising. This is a survey paper, and its prime goal is to provide a broad view of the history of the field of image denoising and closely related topics in image processing. Our aim is to give a better context to recent discoveries, and to the influence of the AI revolution in our domain.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134983002","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}
We present an extension of the linear sampling method for solving the sound-soft inverse acoustic scattering problem with randomly distributed point sources. The theoretical justification of our sampling method is based on the Helmholtz–Kirchhoff identity, the cross-correlation between measurements, and the volume and imaginary near-field operators, which we introduce and analyze. Implementations in MATLAB using boundary elements, the SVD, Tikhonov regularization, and Morozov’s discrepancy principle are also discussed. We demonstrate the robustness and accuracy of our algorithms with several numerical experiments in two dimensions.
{"title":"The Linear Sampling Method for Random Sources","authors":"Josselin Garnier, Houssem Haddar, Hadrien Montanelli","doi":"10.1137/22m1531336","DOIUrl":"https://doi.org/10.1137/22m1531336","url":null,"abstract":"We present an extension of the linear sampling method for solving the sound-soft inverse acoustic scattering problem with randomly distributed point sources. The theoretical justification of our sampling method is based on the Helmholtz–Kirchhoff identity, the cross-correlation between measurements, and the volume and imaginary near-field operators, which we introduce and analyze. Implementations in MATLAB using boundary elements, the SVD, Tikhonov regularization, and Morozov’s discrepancy principle are also discussed. We demonstrate the robustness and accuracy of our algorithms with several numerical experiments in two dimensions.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135520275","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}
We propose a multifrequency algorithm for recovering partial information on the trajectory of a moving point source from one and sparse far-field observation directions in the frequency domain. The starting and terminal time points of the moving source are both supposed to be known. We introduce the concept of observable directions (angles) in the far-field region and derive all observable directions (angles) for straight and circular motions. The existence of nonobservable directions makes this paper much different from inverse stationary source problems. At an observable direction, it is verified that the smallest strip containing the trajectory and perpendicular to the direction can be imaged, provided the angle between the observation direction and the velocity vector of the moving source lies in . If otherwise, one can only expect to recover a strip thinner than this smallest strip for straight and circular motions. The far-field data measured at sparse observable directions can be used to recover the -convex domain of the trajectory. Both two- and three-dimensional numerical examples are implemented to show effectiveness and feasibility of the approach.
{"title":"Imaging a Moving Point Source from Multifrequency Data Measured at One and Sparse Observation Directions (Part I): Far-Field Case","authors":"Hongxia Guo, Guanghui Hu, Guanqiu Ma","doi":"10.1137/23m1545045","DOIUrl":"https://doi.org/10.1137/23m1545045","url":null,"abstract":"We propose a multifrequency algorithm for recovering partial information on the trajectory of a moving point source from one and sparse far-field observation directions in the frequency domain. The starting and terminal time points of the moving source are both supposed to be known. We introduce the concept of observable directions (angles) in the far-field region and derive all observable directions (angles) for straight and circular motions. The existence of nonobservable directions makes this paper much different from inverse stationary source problems. At an observable direction, it is verified that the smallest strip containing the trajectory and perpendicular to the direction can be imaged, provided the angle between the observation direction and the velocity vector of the moving source lies in . If otherwise, one can only expect to recover a strip thinner than this smallest strip for straight and circular motions. The far-field data measured at sparse observable directions can be used to recover the -convex domain of the trajectory. Both two- and three-dimensional numerical examples are implemented to show effectiveness and feasibility of the approach.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136272285","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}
Photoacoustic tomography (PAT) is a novel and promising technology in hybrid medical imaging that involves generating acoustic waves in the object of interest by stimulating electromagnetic energy. The acoustic wave is measured outside the object. One of the key mathematical problems in PAT is the reconstruction of the initial function that contains diagnostic information from the solution of the wave equation on the surface of the acoustic transducers. Herein, we propose a wave forward operator that assigns an initial function to obtain the solution of the wave equation on a unit sphere. Under the assumption of the radial variable speed of ultrasound, we obtain the singular value decomposition of this wave forward operator by determining the orthonormal basis of a certain Hilbert space comprising eigenfunctions. In addition, numerical simulation results obtained using the continuous Galerkin method are utilized to validate the inversion resulting from the singular value decomposition.
{"title":"Singular Value Decomposition of the Wave Forward Operator with Radial Variable Coefficients","authors":"Minam Moon, Injo Hur, Sunghwan Moon","doi":"10.1137/22m1511643","DOIUrl":"https://doi.org/10.1137/22m1511643","url":null,"abstract":"Photoacoustic tomography (PAT) is a novel and promising technology in hybrid medical imaging that involves generating acoustic waves in the object of interest by stimulating electromagnetic energy. The acoustic wave is measured outside the object. One of the key mathematical problems in PAT is the reconstruction of the initial function that contains diagnostic information from the solution of the wave equation on the surface of the acoustic transducers. Herein, we propose a wave forward operator that assigns an initial function to obtain the solution of the wave equation on a unit sphere. Under the assumption of the radial variable speed of ultrasound, we obtain the singular value decomposition of this wave forward operator by determining the orthonormal basis of a certain Hilbert space comprising eigenfunctions. In addition, numerical simulation results obtained using the continuous Galerkin method are utilized to validate the inversion resulting from the singular value decomposition.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135442011","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}
Shuai Huang, Mona Zehni, Ivan Dokmanić, Zhizhen Zhao
Unknown view tomography (UVT) reconstructs a 3D density map from its 2D projections at unknown, random orientations. A line of work starting with Kam (1980) employs the method of moments with rotation-invariant Fourier features to solve UVT in the frequency domain, assuming that the orientations are uniformly distributed. This line of work includes the recent orthogonal matrix retrieval (OMR) approaches based on matrix factorization, which, while elegant, either require side information about the density that is not available or fail to be sufficiently robust. For OMR to break free from those restrictions, we propose to jointly recover the density map and the orthogonal matrices by requiring that they be mutually consistent. We regularize the resulting nonconvex optimization problem by a denoised reference projection and a nonnegativity constraint. This is enabled by the new closed-form expressions for spatial autocorrelation features. Further, we design an easy-to-compute initial density map which effectively mitigates the nonconvexity of the reconstruction problem. Experimental results show that the proposed OMR with spatial consensus is more robust and performs significantly better than the previous state-of-the-art OMR approach in the typical low signal-to-noise-ratio scenario of 3D UVT.
{"title":"Orthogonal Matrix Retrieval with Spatial Consensus for 3D Unknown View Tomography","authors":"Shuai Huang, Mona Zehni, Ivan Dokmanić, Zhizhen Zhao","doi":"10.1137/22m1498218","DOIUrl":"https://doi.org/10.1137/22m1498218","url":null,"abstract":"Unknown view tomography (UVT) reconstructs a 3D density map from its 2D projections at unknown, random orientations. A line of work starting with Kam (1980) employs the method of moments with rotation-invariant Fourier features to solve UVT in the frequency domain, assuming that the orientations are uniformly distributed. This line of work includes the recent orthogonal matrix retrieval (OMR) approaches based on matrix factorization, which, while elegant, either require side information about the density that is not available or fail to be sufficiently robust. For OMR to break free from those restrictions, we propose to jointly recover the density map and the orthogonal matrices by requiring that they be mutually consistent. We regularize the resulting nonconvex optimization problem by a denoised reference projection and a nonnegativity constraint. This is enabled by the new closed-form expressions for spatial autocorrelation features. Further, we design an easy-to-compute initial density map which effectively mitigates the nonconvexity of the reconstruction problem. Experimental results show that the proposed OMR with spatial consensus is more robust and performs significantly better than the previous state-of-the-art OMR approach in the typical low signal-to-noise-ratio scenario of 3D UVT.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135746488","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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