Identification of physical properties of materials is very important because they are in general unknown. Furthermore, their direct experimental measurement could be costly and inaccurate. In such a situation, a cheap and efficient alternative is to mathematically formulate an inverse, but difficult, problem that can be solved, in general, numerically; the challenge being that the problem is, in general, nonlinear and ill-posed. In this paper, the reconstruction of a lower-order unknown time-dependent coefficient in a Cahn–Hilliard-type fourth-order equation from an additional integral observation, which has application to characterizing the nonlinear saturation of the collisional trapped-ion mode in a tokamak, is investigated. The local existence and uniqueness of the solution to such inverse problem is established by utilizing the Rothe method. Moreover, the continuous dependence of the unknown coefficient upon the measured data is derived. Next, the Tikhonov regularization method is applied to recover the unknown coefficient from noisy measurements. The stability estimate of the minimizer is derived by investigating an auxiliary linear fourth-order inverse source problem. Henceforth, the variational source condition can be verified. Then the convergence rate is obtained under such source condition.
{"title":"Determination of the time-dependent effective ion collision frequency from an integral observation","authors":"Kai Cao, Daniel Lesnic","doi":"10.1515/jiip-2023-0024","DOIUrl":"https://doi.org/10.1515/jiip-2023-0024","url":null,"abstract":"Identification of physical properties of materials is very important because they are in general unknown. Furthermore, their direct experimental measurement could be costly and inaccurate. In such a situation, a cheap and efficient alternative is to mathematically formulate an inverse, but difficult, problem that can be solved, in general, numerically; the challenge being that the problem is, in general, nonlinear and ill-posed. In this paper, the reconstruction of a lower-order unknown time-dependent coefficient in a Cahn–Hilliard-type fourth-order equation from an additional integral observation, which has application to characterizing the nonlinear saturation of the collisional trapped-ion mode in a tokamak, is investigated. The local existence and uniqueness of the solution to such inverse problem is established by utilizing the Rothe method. Moreover, the continuous dependence of the unknown coefficient upon the measured data is derived. Next, the Tikhonov regularization method is applied to recover the unknown coefficient from noisy measurements. The stability estimate of the minimizer is derived by investigating an auxiliary linear fourth-order inverse source problem. Henceforth, the variational source condition can be verified. Then the convergence rate is obtained under such source condition.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"4 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider time-harmonic electromagnetic wave scattering by an infinitely long, cylindrical, orthotropic dielectric partially coated with a very thin layer of a highly conductive material, which can be modeled by a transmission problem with mixed boundary conditions. Having established the well-posedness of the direct and interior transmission problem by the variational method under certain conditions, we make use of the classical linear sampling method to reconstruct the shape of the obstacle. Then, based on a modification of the general data-to-pattern operator G, we propose a novel and simple method to justify the modified linear sampling method.
考虑时谐电磁波在无限长、圆柱形、正交电介质上的散射，电介质上部分涂有一层非常薄的高导电性材料，可以用具有混合边界条件的传输问题来模拟。在一定条件下，我们利用变分法确定了直接和内部传输问题的良好求解性，并利用经典线性采样法重建了障碍物的形状。然后，基于对一般数据到模式算子 G 的修正，我们提出了一种新颖而简单的方法来论证修正的线性采样法。
{"title":"Modified linear sampling method for inverse scattering by a partially coated dielectric","authors":"Jianli Xiang, Guozheng Yan","doi":"10.1515/jiip-2022-0096","DOIUrl":"https://doi.org/10.1515/jiip-2022-0096","url":null,"abstract":"Consider time-harmonic electromagnetic wave scattering by an infinitely long, cylindrical, orthotropic dielectric partially coated with a very thin layer of a highly conductive material, which can be modeled by a transmission problem with mixed boundary conditions. Having established the well-posedness of the direct and interior transmission problem by the variational method under certain conditions, we make use of the classical linear sampling method to reconstruct the shape of the obstacle. Then, based on a modification of the general data-to-pattern operator <jats:italic>G</jats:italic>, we propose a novel and simple method to justify the modified linear sampling method.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"303 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The inverse Sturm–Liouville problem with smooth potentials is considered. The high-order estimate of the entire function associated with two Sturm–Liouville problems is established. Applying this estimate expression to inverse Sturm–Liouville problems, we proved that the conclusion in [L. Amour, J. Faupin and T. Raoux, Inverse spectral results for Schrödinger operators on the unit interval with partial information given on the potentials, J. Math. Phys. 50 2009, 3, Article ID 033505] remains true for more general case.
研究了具有平滑势的反 Sturm-Liouville 问题。建立了与两个 Sturm-Liouville 问题相关的全函数的高阶估计。将这一估算表达式应用于反 Sturm-Liouville 问题，我们证明了 [L. Amour, J. Faupin and T.Amour, J. Faupin and T. Raoux, Inverse spectral results for Schrödinger operators on the unit interval with partial information given on the potentials, J. Math.50 2009, 3, Article ID 033505] 中的结论对于更一般的情况仍然适用。
{"title":"The high-order estimate of the entire function associated with inverse Sturm–Liouville problems","authors":"Zhaoying Wei, Guangsheng Wei, Yan Wang","doi":"10.1515/jiip-2023-0082","DOIUrl":"https://doi.org/10.1515/jiip-2023-0082","url":null,"abstract":"The inverse Sturm–Liouville problem with smooth potentials is considered. The high-order estimate of the entire function associated with two Sturm–Liouville problems is established. Applying this estimate expression to inverse Sturm–Liouville problems, we proved that the conclusion in [L. Amour, J. Faupin and T. Raoux, Inverse spectral results for Schrödinger operators on the unit interval with partial information given on the potentials, J. Math. Phys. 50 2009, 3, Article ID 033505] remains true for more general case.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"21 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139644793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In X-ray CT imaging, there are some cases where the obtained CT images have serious ring artifacts and noise, and these degraded CT images seriously affect the quality of clinical diagnosis. Thus, developing an effective method that can simultaneously suppress ring artifacts and noise is of great importance. Total variation (TV) is a famous prior regularization for image denoising in the image processing field, however, for degraded CT images, it can suppress the noise but fail to reduce the ring artifacts. To address this issue, the L0L_{0} smoothing filter is incorporated with TV prior for CT ring artifacts and noise removal problem where the problem is transformed into several optimization sub-problems which are iteratively solved. The experiments demonstrate that the ring artifacts and noise presented in the CT image can be effectively suppressed by the proposed method and meanwhile the detailed features such as edge structure can be well preserved. As the superiority of TV and L0L_{0} smoothing filters are fully utilized, the performance of the proposed method is better than the existing methods such as the TV-based method and L0L_{0}-based method.
在 X 射线 CT 成像中，有些情况下获得的 CT 图像存在严重的环状伪影和噪声，这些劣化的 CT 图像严重影响了临床诊断的质量。因此，开发一种能同时抑制环状伪影和噪声的有效方法就显得尤为重要。全变异（TV）是图像处理领域中一种著名的图像去噪先验正则化方法，但对于劣化的 CT 图像，它能抑制噪声，却无法减少环状伪影。为了解决这个问题，我们将 L 0 L_{0} 平滑滤波器与 TV 先验相结合，用于 CT 环状伪影和噪声去除问题，并将问题转化为多个优化子问题，通过迭代求解。实验证明，该方法能有效抑制 CT 图像中出现的环状伪影和噪声，同时还能很好地保留边缘结构等细节特征。由于充分发挥了电视滤波器和 L 0 L_{0} 平滑滤波器的优势，所提方法的性能优于基于电视的方法和基于 L 0 L_{0} 的方法等现有方法。
{"title":"CT image restoration method via total variation and L 0 smoothing filter","authors":"Hai Yin, Xianyun Li, Zhi Liu, Wei Peng, Chengxiang Wang, Wei Yu","doi":"10.1515/jiip-2023-0052","DOIUrl":"https://doi.org/10.1515/jiip-2023-0052","url":null,"abstract":"In X-ray CT imaging, there are some cases where the obtained CT images have serious ring artifacts and noise, and these degraded CT images seriously affect the quality of clinical diagnosis. Thus, developing an effective method that can simultaneously suppress ring artifacts and noise is of great importance. Total variation (TV) is a famous prior regularization for image denoising in the image processing field, however, for degraded CT images, it can suppress the noise but fail to reduce the ring artifacts. To address this issue, the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>L</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jiip-2023-0052_eq_0016.png\" /> <jats:tex-math>L_{0}</jats:tex-math> </jats:alternatives> </jats:inline-formula> smoothing filter is incorporated with TV prior for CT ring artifacts and noise removal problem where the problem is transformed into several optimization sub-problems which are iteratively solved. The experiments demonstrate that the ring artifacts and noise presented in the CT image can be effectively suppressed by the proposed method and meanwhile the detailed features such as edge structure can be well preserved. As the superiority of TV and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>L</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jiip-2023-0052_eq_0016.png\" /> <jats:tex-math>L_{0}</jats:tex-math> </jats:alternatives> </jats:inline-formula> smoothing filters are fully utilized, the performance of the proposed method is better than the existing methods such as the TV-based method and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>L</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jiip-2023-0052_eq_0016.png\" /> <jats:tex-math>L_{0}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-based method.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"288 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose an inverse problem of determining the mechanical and variable-order parameters of the Euler–Bernoulli beam on viscoelastic foundation. For this goal, we develop a fully-discrete Hermite finite element scheme for this model and analyze the corresponding error estimates. The Levenberg–Marquardt method is then applied to determine the multiple parameters. Extensive numerical experiments are performed under practical settings to demonstrate the behavior of the proposed model and the efficiency of the algorithm.
{"title":"Inverting mechanical and variable-order parameters of the Euler–Bernoulli beam on viscoelastic foundation","authors":"Jin Cheng, Zhiwei Yang, Xiangcheng Zheng","doi":"10.1515/jiip-2023-0084","DOIUrl":"https://doi.org/10.1515/jiip-2023-0084","url":null,"abstract":"We propose an inverse problem of determining the mechanical and variable-order parameters of the Euler–Bernoulli beam on viscoelastic foundation. For this goal, we develop a fully-discrete Hermite finite element scheme for this model and analyze the corresponding error estimates. The Levenberg–Marquardt method is then applied to determine the multiple parameters. Extensive numerical experiments are performed under practical settings to demonstrate the behavior of the proposed model and the efficiency of the algorithm.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"172 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139644878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper is devoted to a problem of acquiring elastic properties of a composite material from the vibration testing data with a simplified experimental acquisition scheme. The specimen is considered to abide by the linear elasticity laws and subject to viscoelastic damping. The boundary value problem for transverse movement of such a specimen in the frequency domain is formulated and solved with finite-element method. The correction method is suggested for the finite element matrices to account for the mass of the accelerometer. The problem of acquiring the elastic parameters is then formulated as a nonlinear least-square optimization problem. The usage of the automatic differentiation technique for stable and efficient computation of the gradient and hessian allows to use well-studied first and second order local optimization methods. We also explore the possibility of generating initial guesses for local minimization by heuristic global methods. The results of the numerical experiments on simulated data are analyzed in order to provide insights for the following real life experiments.
{"title":"Acquiring elastic properties of thin composite structure from vibrational testing data","authors":"Vitalii Aksenov, Alexey Vasyukov, Katerina Beklemysheva","doi":"10.1515/jiip-2022-0081","DOIUrl":"https://doi.org/10.1515/jiip-2022-0081","url":null,"abstract":"The paper is devoted to a problem of acquiring elastic properties of a composite material from the vibration testing data with a simplified experimental acquisition scheme. The specimen is considered to abide by the linear elasticity laws and subject to viscoelastic damping. The boundary value problem for transverse movement of such a specimen in the frequency domain is formulated and solved with finite-element method. The correction method is suggested for the finite element matrices to account for the mass of the accelerometer. The problem of acquiring the elastic parameters is then formulated as a nonlinear least-square optimization problem. The usage of the automatic differentiation technique for stable and efficient computation of the gradient and hessian allows to use well-studied first and second order local optimization methods. We also explore the possibility of generating initial guesses for local minimization by heuristic global methods. The results of the numerical experiments on simulated data are analyzed in order to provide insights for the following real life experiments.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"9 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139474800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper is concerned with discontinuous inverse problem generated by complex-valued weight Sturm–Liouville differential operator with λ-dependent boundary conditions. We establish some properties of spectral characteristic and prove that the potential on the whole interval can be uniquely determined by the Weyl-type function or two spectra.
{"title":"Inverse problem for Sturm–Liouville operator with complex-valued weight and eigenparameter dependent boundary conditions","authors":"Gaofeng Du, Chenghua Gao","doi":"10.1515/jiip-2023-0081","DOIUrl":"https://doi.org/10.1515/jiip-2023-0081","url":null,"abstract":"Abstract This paper is concerned with discontinuous inverse problem generated by complex-valued weight Sturm–Liouville differential operator with λ-dependent boundary conditions. We establish some properties of spectral characteristic and prove that the potential on the whole interval can be uniquely determined by the Weyl-type function or two spectra.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"1 11","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139437914","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Alexander E. Chernyavsky, Leonid L. Frumin, Andrey A. Gelash
We consider right and left formulations of the inverse scattering problem for the Zakharov–Shabat system and the corresponding integral Gelfand–Levitan–Marchenko equations. Both formulations are helpful for numerical solving of the inverse scattering problem, which we perform using the previously developed Toeplitz Inner Bordering (TIB) algorithm. First, we establish general relations between the right and left scattering coefficients. Then we propose an auxiliary kernel of the left Gelfand–Levitan–Marchenko equations, which allows one to solve the right scattering problem numerically. We generalize the TIB algorithm, initially proposed in the left formulation, to the right scattering problem case with the obtained formulas. The test runs of the TIB algorithm illustrate our results reconstructing the various non-symmetrical potentials from their right scattering data.
{"title":"Right and left inverse scattering problems formulations for the Zakharov–Shabat system","authors":"Alexander E. Chernyavsky, Leonid L. Frumin, Andrey A. Gelash","doi":"10.1515/jiip-2022-0087","DOIUrl":"https://doi.org/10.1515/jiip-2022-0087","url":null,"abstract":"We consider right and left formulations of the inverse scattering problem for the Zakharov–Shabat system and the corresponding integral Gelfand–Levitan–Marchenko equations. Both formulations are helpful for numerical solving of the inverse scattering problem, which we perform using the previously developed Toeplitz Inner Bordering (TIB) algorithm. First, we establish general relations between the right and left scattering coefficients. Then we propose an auxiliary kernel of the left Gelfand–Levitan–Marchenko equations, which allows one to solve the right scattering problem numerically. We generalize the TIB algorithm, initially proposed in the left formulation, to the right scattering problem case with the obtained formulas. The test runs of the TIB algorithm illustrate our results reconstructing the various non-symmetrical potentials from their right scattering data.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"53 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139423279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose two parameter choice rules for the discretizing Tikhonov regularization via multiscale Galerkin projection for solving linear ill-posed integral equations. In contrast to previous theoretical analyses, we introduce a new concept called the projection noise level to obtain error estimates for the approximate solutions. This concept allows us to assess how noise levels change during projection. The balance principle and Hanke–Raus rule are modified by incorporating the error estimates of the projection noise level. We demonstrate the convergence rate of these two modified parameter choice rules through rigorous proof. In addition, we find that the error between the approximate solution and the exact solution improves as the noise frequency increases. Finally, numerical experiments are provided to illustrate the theoretical findings presented in this paper.
{"title":"A discretizing Tikhonov regularization method via modified parameter choice rules","authors":"Rong Zhang, Feiping Xie, Xingjun Luo","doi":"10.1515/jiip-2023-0056","DOIUrl":"https://doi.org/10.1515/jiip-2023-0056","url":null,"abstract":"In this paper, we propose two parameter choice rules for the discretizing Tikhonov regularization via multiscale Galerkin projection for solving linear ill-posed integral equations. In contrast to previous theoretical analyses, we introduce a new concept called the projection noise level to obtain error estimates for the approximate solutions. This concept allows us to assess how noise levels change during projection. The balance principle and Hanke–Raus rule are modified by incorporating the error estimates of the projection noise level. We demonstrate the convergence rate of these two modified parameter choice rules through rigorous proof. In addition, we find that the error between the approximate solution and the exact solution improves as the noise frequency increases. Finally, numerical experiments are provided to illustrate the theoretical findings presented in this paper.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"81 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139374225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Electrical Impedance Tomography (EIT) is a widely employed imaging technique in industrial inspection, geophysical prospecting, and medical imaging. However, the inherent nonlinearity and ill-posedness of EIT image reconstruction present challenges for classical regularization techniques, such as the critical selection of regularization terms and the lack of prior knowledge. Deep generative models (DGMs) have been shown to play a crucial role in learning implicit regularizers and prior knowledge. This study aims to investigate the potential of three DGMs – variational autoencoder networks, normalizing flow, and score-based diffusion model – to learn implicit regularizers in learning-based EIT imaging. We first introduce background information on EIT imaging and its inverse problem formulation. Next, we propose three algorithms for performing EIT inverse problems based on corresponding DGMs. Finally, we present numerical and visual experiments, which reveal that (1) no single method consistently outperforms the others across all settings, and (2) when reconstructing an object with two anomalies using a well-trained model based on a training dataset containing four anomalies, the conditional normalizing flow (CNF) model exhibits the best generalization in low-level noise, while the conditional score-based diffusion model (CSD*) demonstrates the best generalization in high-level noise settings. We hope our preliminary efforts will encourage other researchers to assess their DGMs in EIT and other nonlinear inverse problems.
电阻抗断层扫描（EIT）是一种广泛应用于工业检测、地球物理勘探和医学成像的成像技术。然而，EIT 图像重构固有的非线性和非假设性给经典的正则化技术带来了挑战，如正则化项的关键选择和先验知识的缺乏。深度生成模型（DGM）已被证明在学习隐式正则化器和先验知识方面发挥着至关重要的作用。本研究旨在研究三种 DGM（变异自动编码器网络、归一化流和基于分数的扩散模型）在基于学习的 EIT 成像中学习隐式正则化器的潜力。我们首先介绍了 EIT 成像的背景信息及其逆问题的表述。接下来，我们提出了三种基于相应 DGM 的 EIT 反问题算法。最后，我们介绍了数值和视觉实验，实验结果表明：(1) 在所有情况下，没有一种方法能始终优于其他方法；(2) 当使用基于包含四个异常点的训练数据集的训练有素的模型重建有两个异常点的物体时，条件归一化流模型（CNF）在低水平噪声中表现出最佳泛化效果，而基于条件分数的扩散模型（CSD*）在高水平噪声中表现出最佳泛化效果。我们希望我们的初步努力能鼓励其他研究人员在 EIT 和其他非线性逆问题中评估他们的 DGM。
{"title":"A comparative study of variational autoencoders, normalizing flows, and score-based diffusion models for electrical impedance tomography","authors":"Huihui Wang, Guixian Xu, Qingping Zhou","doi":"10.1515/jiip-2023-0037","DOIUrl":"https://doi.org/10.1515/jiip-2023-0037","url":null,"abstract":"Electrical Impedance Tomography (EIT) is a widely employed imaging technique in industrial inspection, geophysical prospecting, and medical imaging. However, the inherent nonlinearity and ill-posedness of EIT image reconstruction present challenges for classical regularization techniques, such as the critical selection of regularization terms and the lack of prior knowledge. Deep generative models (DGMs) have been shown to play a crucial role in learning implicit regularizers and prior knowledge. This study aims to investigate the potential of three DGMs – variational autoencoder networks, normalizing flow, and score-based diffusion model – to learn implicit regularizers in learning-based EIT imaging. We first introduce background information on EIT imaging and its inverse problem formulation. Next, we propose three algorithms for performing EIT inverse problems based on corresponding DGMs. Finally, we present numerical and visual experiments, which reveal that (1) no single method consistently outperforms the others across all settings, and (2) when reconstructing an object with two anomalies using a well-trained model based on a training dataset containing four anomalies, the conditional normalizing flow (CNF) model exhibits the best generalization in low-level noise, while the conditional score-based diffusion model (CSD*) demonstrates the best generalization in high-level noise settings. We hope our preliminary efforts will encourage other researchers to assess their DGMs in EIT and other nonlinear inverse problems.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"85 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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